Russia's military pounded residential areas across Ukraine overnight as Ukrainian forces pressed a counteroffensive to try to take back an occupied southern region, striking the last working bridge over a river in the Russian-occupied Kherson region, Ukrainian authorities said Saturday.

A Russian rocket attack on Kramatorsk killed three people and wounded 13 on Friday night, according to the mayor. The city is the headquarters for Ukrainian forces in the war-torn east.

The attack came less than a day after 11 other rockets were fired at the city, one of the two main Ukrainian-held ones in Donetsk province, the focus of a Russian offensive to capture Eastern Ukraine's Donbas region

The Russian Defence Ministry claimed Saturday its forces had taken control of Pisky, a village on the outskirts of the city of Donetsk, the provincial capital that pro-Moscow separatists have claimed since 2014.

But Ukraine's military command later said in a Facebook post that "fierce fighting continues" in the area.

Russian troops and the Kremlin-backed rebels are seeking to seize Ukrainian-held areas north and west of the city of Donetsk to expand the separatists' self-proclaimed republic.

But the Ukrainian military said Saturday its forces had prevented an overnight advance toward the smaller cities of Avdiivka and Bakhmut.

Russian Defence Ministry spokesperson Igor Konashenkov also claimed that Russian strikes near Kramatorsk, 120 kilometres north of Donetsk city, destroyed a U.S.-supplied multiple rocket launcher and ammunition. Ukrainian authorities didn't acknowledge any military losses, but said Russian missile strikes Friday on Kramatorsk had destroyed 20 residential buildings.

Neither claim could be independently verified.