After weeks of unexpected battlefield setbacks for Russia, the war in Ukraine on Sunday delivered another surprise: the emergence of a former Russian convict and onetime hot-dog seller as perhaps the Kremlin’s best hope for a small, face-saving military victory.

With occupying Russian forces at peril in the strategic southern city of Kherson, troops with a private military force controlled by Yevgeny V. Prigozhin, a convicted thief and longtime associate of Russia’s president, Vladimir Putin, advanced on the Ukrainian-held city of Bakhmut in the country’s east.

The city, under attack by Russia for months, has little strategic value, but a victory there for Moscow would break its humiliating run of defeats — and give a boost to the political fortunes of Prigozhin, a shadowy businessman who served nine years in a Soviet prison for robbery. Prigozhin used to be mocked as "Putin’s cook” because of his business interests in catering but is now a growing force in Russia’s labyrinthine power politics.