U.S. officials are still trying to learn whether the general, Sergei Surovikin, a former top commander of Moscow’s forces in Ukraine who holds significant support in the Russian military, helped plan the mutiny led by Yevgeny V. Prigozhin, the Wagner mercenary group’s leader. If General Surovikin is determined to have been involved, it will pose serious questions for Mr. Putin about how to respond.

Here are other developments:

• Rescuers combed through rubble in the eastern Ukrainian city of Kramatorsk on Wednesday as the death toll from a Russian missile strike on a popular restaurant climbed to 10, the Ukrainian police said. Dozens of people were injured in the latest major strike on the city, which has suffered several deadly attacks by Russian forces during the war.

• Three days after calling off his Wagner forces’ advance on Moscow, Mr. Prigozhin arrived in neighboring Belarus on Tuesday as part of an agreement that secured his amnesty in exchange for exile. President Aleksandr G. Lukashenko of Belarus said that Mr. Putin had entertained the possibility of killing Mr. Prigozhin. But the Belarusian leader, a loyal Putin ally, said he had talked Mr. Putin out of doing so, while also warning Mr. Prigozhin that Mr. Putin could “squash him like a bug.”

• The Pentagon said it was sending an additional $500 million in weapons to Ukraine, including 55 Bradley and Stryker armored vehicles, and equipment for clearing minefields.

• The authorities in Kyiv announced a new, high-tech bomb shelter system, including automatic doors and digital keypads, as officials in the Ukrainian capital sought to address criticism after three people were killed this month by a missile at the doorway of a locked shelter.

With search and rescue efforts ongoing, Ukraine’s intelligence agency, the S.B.U., said it had detained a man who allegedly helped direct the missile strike. It accused the man — described as a Kramatorsk resident working for a local gas transportation company — of sending video footage of the restaurant to Russian military intelligence. The S.B.U. did not name the person and the accusations could not be independently verified.

The restaurant’s terrace was a mess of overturned black-and-white sofas on Wednesday morning. As a crane lifted debris off a large pile of twisted rubble, distraught friends and relatives of people still missing waited anxiously behind police ribbon across the street. The restaurant’s owner paced while talking on her phone.

One woman was awaiting news of her niece. A man was seeking information about his brother-in-law. When emergency workers came out to take breaks, pouring water onto their heads to wash off the dust, people rushed over to ask them for updates — with one man showing a photo of his missing relative on his phone.

As two more bodies were pulled from the rubble, people strained to get a glimpse to see whether they were their loved ones. But the body bags were zipped tightly, and an emergency car quickly took them to the morgue.

Ria Lounge, known to many as Ria Pizza, was a long-running haunt in Kramatorsk and was particularly popular in the summer because of its covered outdoor seating. The restaurant, which was on the ground floor, had closed after Russia’s full-scale invasion of Ukraine began in February last year but reopened several months later. It is close to the Hotel Kramatorsk, which was badly damaged in a Russian attack last summer.

Ukrainian soldiers stationed nearby, some of them newly returned from the front, are frequent patrons, along with local residents, foreign journalists and aid workers.

Petro Golub, 66, said he had been sitting on a bench near the restaurant when he heard what sounded like a plane before an explosion. The blast wave threw him “all the way back,” Mr. Golub said.

“I am fine, fine, just little things — my legs were slashed by glass shards,” he said on Wednesday.

Kramatorsk, in the Donetsk region of eastern Ukraine, was home to 150,000 people before the war, but many have fled as Russia has set its sights on taking the whole region. It is far enough away from the front line for life to seem relatively normal on most days, but air raid sirens are common, and the sounds of distant artillery are sometimes heard.

It has been hit with a devastating attack before: A Russian missile strike hit a packed train station in April, killing more than 50 people. And the city featured heavily in Russia’s monthslong campaign to take the city of Bakhmut, serving as a way station for Ukrainian troops rotating out of the grueling fight.

Bakhmut’s capture by Russian forces in May left Kramatorsk under a greater threat and facing an increased risk of shelling. Ukrainian officials had also warned that Bakhmut’s fall could have a domino effect, clearing the way for Russian forces to advance on Kramatorsk, the city of Sloviansk and the rest of Donetsk.

Addressing a New York Times report that Gen. Sergei Surovikin had prior knowledge of last weekend’s armed rebellion by Yevgeny Prigozhin, Kremlin spokesman Dmitri Peskov said: “There will be a lot of various speculations, gossip and so on around these events — I think this is one of such examples.” Peskov insisted that Russia had unified around Vladimir Putin, saying: “The army, the people, everyone was next to the president.”

It’s notable that Peskov’s curt response, as carried by the Russian state news media, did not include an outright denial of The Times’s reporting about Surovikin or an expression of the Kremlin’s confidence in the general, who is a former commander of Russian forces in Ukraine.

“There are now many ways available,” said Mr. Klitschko, who was reprimanded this week in connection with the shelter problems. “People can get into these premises as quickly as possible.”

Kyiv residents were deeply unsettled after a woman, her daughter and another woman died in a Russian missile strike on June 1 outside the doorway of a locked shelter that they were desperately trying to get into. Missile fragments struck them as they waited for someone to open the door.

The police have opened a criminal investigation, and four people have been questioned on suspicion of official negligence, a serious charge that can carry sentences of up to five years in prison. The shelter’s guard remains in jail.

The episode uncapped a litany of complaints about bomb shelters, which are on many people’s minds in a city that is relentlessly hit by Russian drones and missiles. Residents said that many shelters were always closed, that others were in deplorable condition and that it was difficult to figure out which shelter to run to in the event of an air raid.

On Tuesday, the administration of President Volodymyr Zelensky officially reprimanded Mr. Klitschko and half a dozen other officials over the shelter problems. Several midlevel administrators were dismissed.

But analysts say that Mr. Klitschko, a celebrated former heavyweight boxing champion, remains too powerful for Mr. Zelensky to remove. The mayor also remains popular for the work he did helping organize life in Kyiv during the war’s most dangerous days last year.

The death toll from a strike on a popular restaurant in Kramatorsk on Tuesday has risen to nine, including three children, according to the Ministry of Internal Affairs, with 56 people still injured in the attack.

The death toll from a Russian missile strike on a crowded restaurant in the eastern Ukrainian city of Kramatorsk has risen to eight, Ukraine’s Emergency Services said. The attack on Tuesday at dinnertime also injured 56 people, it said, adding that rescuers were still searching for people in the rubble.

As rattled as they may have been by an armed insurrection in a nuclear-weapons state, Russia’s friends and business partners are unlikely to abandon Mr. Putin, according to diplomats and analysts. The more likely scenario, they say, is for them to hedge their bets against further Russian instability.

“I’m not surprised at any of those public statements,” said Michael A. McFaul, a former American ambassador to Russia. “It’s not in our interest or anyone else’s interest to stir things up. But privately, if your goal is stability, then you should be worried about Putin’s ability to provide this stability.”

China, Mr. Putin’s most important patron, views Russia as a linchpin in its campaign to blunt the global ambitions of the United States. On Monday, the Chinese Foreign Ministry reaffirmed its support for Russia.

Evan S. Medeiros, a professor of Asian studies at Georgetown University, said, “A weak Russia denies China an ally in its competition with the U.S. and, perhaps worse, leaves Xi isolated globally and under pressure from democracies.”

Saudi Arabia and other gulf countries have overlooked Russia’s war on Ukraine because they increasingly viewed Mr. Putin as an alternative source of security in a volatile region where the United States is viewed as pulling back.

A parade of gulf leaders has called Mr. Putin in recent days, from the emir of Qatar and the president of the United Arab Emirates, to Crown Prince Mohammed bin Salman of Saudi Arabia. The Saudi leader on Tuesday expressed support for the “steps made by Russia to defend the constitutional order,” according to the Kremlin.

That predictable reaction papers over tensions between Saudi Arabia and Russia over Moscow’s decision to sell oil at cut rates even as Saudi Arabia tries to prop up the price.

How Mr. Putin handles the aftermath of the rebellion will also have an effect on perceptions of his standing.

Mr. Putin has so far held off on reprisals, though he has already cracked down on dissent.

For some countries, like Israel, the calculations can be fiendishly complex. An American ally under pressure to back Ukraine, Israel has been loath to antagonize Mr. Putin because of Russia’s military presence in Syria.

But the Kremlin in recent months has been distracted from Syria by the war in Ukraine, giving Israel’s government some space. It was telling that reports emerged in Israel that Prime Minister Benjamin Netanyahu was contemplating a visit to the Ukrainian capital, Kyiv.

There was far less equivocation in Iran, which is tied to Russia by oil, weapons sales and a kindred sense of global isolation. Iran’s president, Ebrahim Raisi, spoke to Mr. Putin on Monday to offer “his full support,” according to a Kremlin readout of the call.

And for other countries, like India, there are major economic consequences to recalibrating relations with Russia. India, which remains neutral in the conflict, has emerged as one of the largest buyers of Russian oil.

Reporting was contributed by Ivan Nechepurenko in Tbilisi, Georgia, Chris Buckley in Taipei, Taiwan, David Pierson in Hong Kong, and Farnaz Fassihi in New York.

The foreign minister of Lithuania, Gabrielius Landsbergis, said on Tuesday that the relocation was creating a “more volatile, unpredictable environment for our region,” in part because of how quickly Wagner forces mobilized during their short-lived rebellion. Aside from Lithuania, Latvia, Ukraine and Russia, which it counts as a close ally, Belarus neighbors Poland, which has already announced increased security measures along their shared border.

“It could take them eight to 10 hours to suddenly appear somewhere in Belarus, somewhere close to Lithuania, somewhere close to Estonian border,” Mr. Landsbergis said.

Speaking alongside Mr. Landsbergis at a joint news conference in Paris, the Latvian foreign minister, Edgars Rinkevics, said his country viewed the Wagner mutiny as an internal Russian matter, echoing comments made on Monday by the NATO secretary general, Jens Stoltenberg.

Still, Mr. Rinkevics stressed that the potential relocation of Wagner fighters to Belarus, as well as the security of NATO’s eastern flank, must be taken seriously and discussed at the alliance’s summit in Vilnius, Lithuania, next month.

The Polish president, Andrzej Duda, on Tuesday called the Wagner developments “a very negative signal” for the region, the Polish news media reported.

A soundless video of Mr. Shoigu visiting military positions was released on Monday morning in what some Kremlin watchers interpreted as a tacit sign of support for him. Some military bloggers were quick to point out that the video appeared to have been shot on Friday, before the armed rebellion led by Yevgeny V. Prigozhin, the founder of the Wagner mercenary group.

Mr. Shoigu was also present on Monday as Mr. Putin convened a meeting of his top security chiefs. Footage played on state television showed him sitting around a table with his head bowed and his hands folded.

On Tuesday, as Mr. Putin praised his security forces in a grandly choreographed speech, Mr. Shoigu was again present, wearing his military uniform. Later, Mr. Shoigu held a meeting with his Cuban counterpart at the National Defense Control Center of Russia.

“In conditions when the United States has been carrying out an illegal and illegitimate trade and economic blockade of Cuba for many decades, we are ready to help the Island of Freedom, lend a shoulder to our Cuban friends,” Mr. Shoigu said, according to the Russian military channel Zvezda TV.

Mr. Shoigu and General Gerasimov are considered trusted allies of Mr. Putin, but in the past months they have largely stayed out of public view and have made only highly choreographed appearances, while Mr. Prigozhin published videos of himself on the front line amid corpses, with explosions booming in the distance.

Mr. Prigozhin has repeatedly and publicly criticized both men and complained that they have caused some of the Russian military’s problems. Other prominent Russian leaders have also criticized Mr. Shoigu and General Gerasimov.

In October, after Russia’s retreat from the Ukrainian city of Lyman, Ramzan Kadyrov, the strongman leader of the southern Russian republic of Chechnya — who controls his own paramilitary force — wrote on the Telegram messaging app that Russia’s top military brass had “covered for” an “incompetent” general who should now be “sent to the front to wash his shame off with blood.”

Andrei Guryulov, a hard-line member of Russia’s Parliament from the ruling United Russia party, disparaged the military leadership around the same time.

“The whole problem is not on the ground, but on the Frunzenskaya embankment, where they still do not understand, and do not take ownership of the situation,” he said, referring to the location of the Defense Ministry. “Until something completely different appears in the General Staff, nothing will change.”

Even the staunch Putin ally Aleksandr Dugin, whose daughter was killed last autumn by a car bomb, called Mr. Putin and President Aleksandr G. Lukashenko of Belarus “heroes,” but without naming them seemed to cast blame on supporters of Mr. Shoigu and General Gerasimov for the Wagner rebellion.

“Those who made this situation possible, who committed it, and who could not prevent it, and when it all began, were unable to adequately respond, must be said goodbye to abruptly,” Mr. Dugin wrote on Telegram on Monday.

Mr. Shoigu, who was a very popular minister of emergency situations before becoming defense minister in 2012, has had a long and friendly relationship with Mr. Putin. Long before the full-scale invasion of Ukraine, the two were regularly photographed hunting, fishing and picking mushrooms. Ahead of Mr. Putin’s birthday in 2019, they vacationed together in the vast Russian taiga, taking long hikes. But he has never served in the military, which has been a cause of resentment among his critics.

General Gerasimov is seen as a consummate military man, though some analysts suggested at the time of his appointment that the Kremlin was looking to streamline military decision-making and had appointed him in the hopes of getting a leader willing to carry out decisions coming directly from the top. He has not spoken in public since the revolt.

Mr. Putin may have kept both men in charge as part of his decades-long efforts to place the sprawling Russian military more under his control.

“It’s a Russian paradox,” said Andrei Soldatov, an expert on the Russian security services.

Mr. Putin “needs someone quite weak and compromised to represent the military politically,” he added, “because what he remembers about the recent rise of history in the last 30 years is that even the most disastrous of wars produce popular generals.”

Oleg Matsnev contributed reporting.

The aid was announced in an email sent to reporters on Tuesday afternoon.

In a briefing at the Pentagon shortly afterward, the Defense Department spokesman, Brig. Gen. Patrick S. Ryder, described the matériel being sent as adding “important capabilities” that would contribute to Ukraine’s counteroffensive. He declined to address whether the 30 Bradleys and 25 Strykers being sent in the new tranche of aid were meant to replace earlier such vehicles damaged or destroyed in combat with Russian forces.

“What I’m not going to do is battle damage assessment for the Russian military from the podium here, so I’m not going to get into Ukrainian capabilities and their status on the battlefield,” General Ryder said. “In the midst of combat, one can expect that equipment will be damaged, that equipment will be destroyed. It is a fight. And it is a hard fight.”

“So by us providing additional armor, capabilities, additional ammunition additional breaching capabilities, that’s all meant to enable them to sustain this fight that they find themselves in,” he added.

The list of items provided by the Pentagon includes more anti-armor missiles and rockets, air-launched missiles for attacking Russian air-defense sites, demolition equipment for “obstacle clearing” and mine-clearing equipment.

General Ryder declined to address questions about the aborted mutiny of Wagner Group forces in Russia on Saturday in detail, calling it “an internal Russian matter.” No U.S. military forces were repositioned as a result of the turmoil, he said, nor did Lloyd J. Austin III, the U.S. secretary of defense, speak with his Russian counterpart over the weekend.

“We’re going to stay focused on giving Ukraine what they need to be successful in defending their country and taking back sovereign territory,” the general said.

The United States is imposing sanctions on four companies — in the Central African Republic, United Arab Emirates and Russia — for their suspected roles in financing the expansion of the Wagner mercenary group through “illicit gold dealings,” the Treasury Department said. Sanctions are also being placed on a person involved with Wagner in Mali.

Brian E. Nelson, the under secretary for terrorism and financial intelligence, said that the U.S. would continue to target the Wagner group’s revenue streams “to degrade its expansion and violence in Africa, Ukraine and anywhere else.”

1/17 Since June 1, even the radio has been turned off in my cell. I can't see other people, and when I do I'm not allowed to talk to them.

— Alexey Navalny (@navalny) June 27, 2023

Two Russian missile strikes have hit Kramatorsk, in eastern Ukraine, with the first striking a restaurant in the city center, according to Andriy Yermak, the head of President Volodymyr Zelensky’s office. He did not specify whether there were casualties, but posted an image of thedestruction on Telegram. The local authorities said search and rescue efforts were underway.

There was a large crowd of people inside a restaurant in Kramatorsk’s city center when a missile hit on Tuesday, the head of the regional military administration, Pavlo Kyrylenko, said on national television, adding that the number of casualties was still being clarified. Video posted by a freelance journalist at the scene showed a number of wounded.