Putin's response to Prigozhin's rebellion is to try to hold on to his fighters

 ISW:

Russian President Vladimir Putin gave a speech on June 26 seeking to persuade as many Wagner fighters and leaders as possible to join the Russian military and continue fighting against Ukraine and to cause individuals most loyal to Wagner Group financier Yevgeny Prigozhin to self-identify. Putin continued to denounce the organizers of the armed rebellion as traitors.[1] Putin thanked Russian society and the Russian security forces for defending Russia’s sovereignty and expressed gratitude to Belarusian President Alexander Lukashenko for brokering negotiations with the perpetrators of the rebellion. Putin did not name Prigozhin specifically, but Putin’s speech leaves little room for any rapprochement with Prigozhin.

Putin stated that Russia’s true enemy is Ukraine and distinguished between the Wagner Group fighters and the armed rebellion’s organizers, presumably Prigozhin and Prigozhin loyalists, and offered Wagner Group fighters three choices. Putin gave the Wagner Group commanders and fighters space to distance themselves from Prigozhin’s armed rebellion, stating that “we know that the overwhelming majority of Wagner Group fighters and commanders are also Russian patriots, devoted to their people and state.”[2] Putin stated that Wagner fighters who seek to continue “serving Russia” can sign a contract with the Russian Ministry of Defense (MoD) or other Russian security services, retire and go home, or go to Belarus (presumably to be with Prigozhin).[3] Putin praised the work of Wagner Group commanders likely in an effort to retain them as the Wagner Group integrates into the MoD. The MoD’s ability to retain as many of Wagner’s current commanders as possible during the integration and subordination process is likely critical to maintaining the Wagner Group’s combat effectiveness and morale.

The Kremlin indicated that Russia aims to retain Wagner forces in order to sustain its operations in Ukraine and other international engagements. Putin could have arrested the Wagner commanders for treason but instead offered to forgive and integrate Wagner forces – which indicates his need for trained and effective manpower. Putin is also likely attempting to finalize the Russian MoD-initiated formalization effort. Russian Foreign Minister Sergei Lavrov reassured his foreign counterparts on June 26 that Wagner will continue operations in Mali and the Central African Republic.[4] Putin’s and Lavrov’s rhetoric supports an ongoing domestic information campaign in Russia to forgive and retain Wagner fighters.[5] Local Russian sources also reported that Wagner employees continue to recruit personnel in St. Petersburg, Yekaterinburg, Novosibirsk, and Tyumen.[6]
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This appears to be an admission that the Russian military by itself is inadequate to sustain his war in Ukraine.  Putin is apparently not willing to admit that his war in Ukraine was a mistake.  His unnecessary war in Ukraine has exposed Russian weakness much more than Ukraine could ever have done had it joined NATO.  I suspect that Ukraine's desire to join NATO has actually increased since Putin's attacks. The fact that Russia needs Wagner troops and other convicts to sustain its operations is more evidence of Russia's military inadequacies.

See, also:

Chaos in Russia is morale booster for Ukraine as it pushes on with early stages of counteroffensive

The armed rebellion against the Russian military may have been over in less than 24 hours, but the disarray within the enemy’s ranks was an unexpected gift and timely morale booster for Ukrainian troops.

The spectacle of Yevgeny Prigozhin’s mutiny in the critical military command and control hub in the Russian city of Rostov-on-Don, and later Russia's scramble to fortify Moscow as troops marched to upend the country’s military leadership was greeted “with applause” by commanders of Ukraine’s Eastern Group of Forces, said its spokesperson, Serhii Cherevatiy.

“Soldiers at the front lines are positive about it,” he said. “Any chaos and disorder on the enemy’s side benefits us.”

A video of well-known Ukrainian drone commander “Magyar” watching the revolt while eating enormous amounts of popcorn went viral. A plethora of gleeful memes mocking Russian leader Vladimir Putin inundated social media, and statement after statement from Ukraine’s top brass described the turmoil as a sure sign of more instability to come.
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And:

 Russia sends barefoot convicts to storm Ukrainian positions near Antonivskyi bridge

 Russia is sending former prisoners from the “Storm Z” units to storm Ukrainian military positions near the Antonivskyi bridge in Kherson, according to Telegram reports on June 26.

The supply of Russian troops in this area is currently at such a level that some of the invaders did not even have shoes, as evidenced by a photo posted on pro-Kremlin Telegram channels.

“Summer cottages near Oleshky are once again under enemy physical control,” reads the comment to the posted photo.

...

And:

 Occupiers drop a chemical projectile on Ukrainian positions, with wind in their faces

Russian invasion forces dropped a chemical projectile on Ukrainian soldiers but with the wind blowing the chemicals back at them, a military spokesperson for the army in the Tauride sector, Valeriy Shershen, said on national television on June 25.

The incident happened near Maryinka in Donetsk Oblast, Shershen said.

"A prohibited chemical projectile was reportedly dropped pm a Ukrainian position, an aerosol munition with a suffocating action. (However), the wind was blowing in the enemy's direction," Shershen said.
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And:

 ‘Every day, the Russians are slowly retreating – and the real counter-offensive is yet to come’

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“Some are professional soldiers, but others are just used as cannon fodder,” he said on Monday, showing a grisly phone video of a battle’s aftermath.

“One night, a load of them came at us and we shot them with big-calibre machine guns – when we saw the bodies the next day, we noticed how poorly equipped they were, with a small amount of ammunition each.”

That sense of pity briefly stirred in him again, as he and his son Bogdan, who serves alongside him, mulled the fall-out of Saturday’s abortive Wagner coup attempt in Russia.

Before the attempted coup, Yaroslav said, the threadbare Wagner soldiers at least had a clear chain of command.

“These guys didn’t really know what they were fighting for before, and who knows what they will do now amid all this chaos,” Yaroslav added.

“As Ukrainians, we will plan to fight as hard as ever, but this coup will affect the morale of the Russian lines, as a lot of blood has been spilt.”
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And:

 Chinese people are watching their idea of a strong, unified Russia get dismantled by Wagner's mutiny, and now they're worrying that China might be next

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On Weibo, China's version of Twitter, the hashtag "Putin accuses Wagner head of treason" received 2.37 billion views in just 24 hours.

Posts discussing Wagner Group chief Yevgeny Prigozhin's mutiny, during which he called upon his 25,000 soldiers to march north to Moscow, surged to the top of Weibo's charts on Sunday, according to data seen by Insider.

The rebellion finally put Moscow's months of internal strife on display for Weibo's users, which stood in stark contrast to the usual Chinese state narrative of a mighty, stable Russia standing up to the West.
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And:

 Putin has been exposed as a 'lame duck,' experts say: 'I have the feeling he is not really running the country'

Yevgeny Prigozhin's rebellion may have abruptly ended before Wagner's forces entered Moscow, but the very fact that a warlord and thousands of his soldiers were able to threaten the capital — and that a foreign leader had to step in to negotiate a peace deal — is a humiliating blow to Russian President Vladimir Putin from which he will never fully recover, experts said in the wake of a remarkable 48 hours.

The mutiny was not, ostensibly, about Putin at all.

In his public remarks, Prigozhin directed his criticism at Russia's military leaders, accusing them of mismanaging the war in Ukraine and undermining his private military company — the culmination of a months-long feud, coming just days before the Ministry of Defense was set to formally take over control of groups such as Wagner.

But, experts say, Russians will not soon forget the sight of mercenaries seizing military installations, capturing a city home to more than one million people, and marching to within striking distance of the Kremlin, prompting emergency "counter-terrorist" measures and a desperate attempt to thwart Wagner's advance by tearing up the road into Moscow.
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And:

 Putin’s power ‘ebbing away’ as Russian president disappears after Wagner rebellion

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The mutiny by the Wagner Group, which saw its leader Yevgeny Prigozhin order his troops to march on Moscow, left the Russian president hiding from public view on Sunday after being forced into an amnesty deal.

Though the astonishing revolt was short-lived – with the fighters withdrawing under the agreement that saw Mr Prigozhin exiled to Belarus – it has raised questions about Mr Putin’s grip on power.

The Russian president has not commented publicly since the deal was struck to de-escalate one of the biggest challenges since he rose to power more than two decades ago. He said he was giving top priority to the conflict in Ukraine in excerpts from an interview aired by state television on Sunday, but it appeared to have been recorded before the rebellion.
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It is not clear that there is an alternative to Putin in Russia at this point.  The military is struggling not only against Ukraine forces but also having trouble with internal controls.