“They are terrorizing the people,” said Tamaz Gambarashvili, the head of the civil-military administration in Vovchansk.

The city was occupied by Russian forces for seven months and then, after a sweeping Ukraine counteroffensive in September forced the Russians to withdraw, the residents suffered a grim winter. Russia unleashed a torrent of daily artillery and mortar strikes as part of its winter offensive across eastern Ukraine.

The latest fighting cut electricity and telephone services, adding to people’s difficulties. The local authorities have focused on providing food and other supplies, including construction materials for damaged houses, to the remaining population.

Two villages that lie even closer to the border than Vovchansk are almost abandoned, the police chief said. There are only two residents left in one of the villages.

The head of the local education department, Lyudmila Madiani, said it was providing online classes for 600 children still in the district, but had to suspend them in the last week because the internet went down. Only four of 21 schools in the district have survived the war undamaged, she said.