On Russia’s state-run television, commentators have enthusiastically promoted the Donbas offensive as a decisive battle that could be a turning point in the war. Many point toward May 9, the commemoration of Russia’s 1945 victory over Nazi Germany, as the date when Mr. Putin could claim a semblance of victory in Ukraine.

“The big battle for the Donbas has already started,” said Yuri Podolyaka, a pro-Russia analyst who publishes military reports on his popular channel on Telegram. “The activity of the Russian artillery and air forces has intensified again.”

On Monday, the head of the regional administration in Luhansk, which is part of the Donbas, said that Russian forces had gained control of the town of Kreminna, adding to territory in the region held by Moscow.

Still, those scattered Russian advances carry less psychological punch than lethal strikes on Lviv, a city that has become a critical gateway to safety for the millions of Ukrainians who have fled westward, trying to escape the worst of the fighting. In late February, it was quickly repurposed from a charming tourist destination into a base of operations for a vast relief effort, serving as a channel for humanitarian supplies, aid workers, foreign fighters making their way to frontline cities, and many foreign journalists.

Hundreds of thousands of displaced people have passed through the city’s train and bus stations. For many others, it is a new — if fleeting — home. Lviv, which had about 720,000 residents before the war began, has since welcomed at least 350,000 people displaced from other parts of the country.

Until Monday, the only direct targets that had been hit in Lviv were a fuel storage site and tank facility in the city’s northeast, hit by several missile strikes about three weeks earlier. Before that, a pair of attacks targeted an airport facility and a military base near Lviv, killing at least 35 people.