Russian troops retreated from Lyman, a Ukrainian city in the Donetsk region, on Friday.
The retreat came one day after Putin claimed the region was part of Russia.
Powerful allies to Putin said Russian military leaders should be punished over the retreat.
Powerful supporters of Russian President Vladimir Putin criticized the military's retreat from a city in Ukraine on Saturday, one day after the president claimed to have annexed the region.
Putin on Friday signed decrees claiming four regions in Ukraine were now part of Russia in a move that was widely rejected by the West as illegal and invalid. However a day later, Ukrainian troops entered the city of Lyman, forcing Russian troops to retreat from an area Putin just declared part of Russia. Lyman is located in the Donetsk region, which Putin claimed to annex.
In a post on Telegram, Ramzan Kadyrov - leader of Chechnya and an ally of Putin's - said Russian military leadership had "covered for" an "incompetent" general that should be "sent to the front to wash his shame off with blood," according to The New York Times translation.
Another prominent supporter and close confidant of Putin, oligarch Yevgeny Prigozhin, echoed those sentiments in a statement. Referring to Russia's military generals, he said, "Send all these pieces of garbage barefoot with machine guns straight to the front," The Times reported.
Russian Defense Minister Sergei K. Shoigu has come under fire recently and been accused of military failings, especially after Ukraine's successful counteroffensives in September. Following Ukraine's advances, Putin escalated the war by mobilizing troops, which also prompted complaints and even protests in Russia.
Putin also appears to be on uncertain ground with key international partners, including China and India. In a United Nations Security Council vote to condemn Putin's "illegal" annexation of Ukraine on Friday, both countries abstained from voting.
Russia was the sole veto vote.