Russia's military pounded residential areas across Ukraine overnight, claiming gains, 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 the city of Kramatorsk killed three people and wounded 13 others Friday night, according to the mayor. Kramatorsk is the headquarters for Ukrainian forces in the country's 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 an ongoing Russian offensive to capture eastern Ukraine's Donbas region.

The Russian Defense 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 controlled since 2014.

Russian troops and the Kremlin-backed rebels are trying 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 that its forces had prevented an overnight advance toward the smaller cities of Avdiivka and Bakhmut.

Russian Defense Ministry spokesman Igor Konashenkov also claimed that Russian strikes near Kramatorsk, 120 kilometers (75 miles) north of Donetsk city, destroyed a U.S.-supplied multiple rocket launcher and ammunition. Ukrainian authorities did not acknowledge any military losses but said Russian missile strikes Friday on Kramatorsk had destroyed 20 residential buildings.

Neither claim could be independently verified.

The Ukrainian governor of neighboring Luhansk province, part of the Donbas region that was overrun by Russian forces last month, claimed that Ukrainian troops still held a small area in the province. Writing on Telegram, Luhansk Gov. Serhii Haidai said the defending troops were holed up inside an oil refinery on the edge of Lysychansk, a city that Moscow claimed to have captured, and also control areas near a village.

“The enemy is burning the ground at the entrances to the Luhansk region because it cannot overcome (Ukrainian resistance along) these few kilometers," Haidai said. "It is difficult to count how many thousands of shells this territory of the free Luhansk region has withstood over the past month and a half.”

Further west, the governor of the Dnipropetrovsk region reported more Russian shelling of the city of Nikopol, which lies across the Dnieper River from Europe's largest nuclear power plant.

Gov. Yevhen Yevtushenko did not specify whether Russian troops had fired at Nikopol from the occupied Zaporizhzhia Nuclear Power Plant. Writing on Telegram, he said Saturday that there were no casualties but residential buildings, a power line and a gas pipeline were damaged.

Nikopol has undergone daily bombardment for most of the past week, and a volley of shells killed three people and damaged 40 apartment buildings on Thursday, he said.

Russia and Ukrainian officials have accused each other of shelling the Zaporizhzhia plant in contravention of nuclear safety rules. Russian troops have occupied the plant since the early days of Moscow's invasion, although the facility's Ukrainian nuclear workers continue to run it.

Read:Russian shelling heavy in east; Ukraine strikes key bridge

Ukrainian military intelligence alleged Saturday that Russian troops were shelling the plant from a village just kilometers away, damaging a plant pumping station and a fire station. The intelligence directorate said the Russians had bused people into the power plant and mounted a Ukrainian flag on a gun on the outskirts of Enerhodar, the city where the plant is located.

“Obviously, it will be used for yet another provocation to accuse the armed forces of Ukraine,” the directorate said, without elaborating.

Ukrainian officials have repeatedly alleged that Russian forces were using the plant as a shield while firing at Ukrainian communities across the river, knowing that Ukrainian forces were unlikely to fire back for fear of triggering a nuclear accident.

They said Russian shelling on Friday night killed one woman and injured two other civilians in the city of Zaporizhzhia. Ukraine's southern Mykolayiv region also said a woman died there in shelling.

For several weeks, Ukraine's military has tried to lay the groundwork for a counteroffensive to reclaim southern Ukraine's Russian-occupied Kherson region. A local Ukrainian official reported Saturday that a Ukrainian strike had damaged the last working bridge over the Dnieper River in the region, further crippling Russian supply lines.

“The Russians no longer have any capability to fully turn over their equipment,” Serhii Khlan, a deputy to the Kherson Regional Council, wrote on Facebook.

The British Defense Ministry said Saturday that damage to bridges across the Dnieper means that “ground resupply for the several thousand Russian troops on the west bank is almost certainly reliant on just two pontoon ferry crossing points.”

“Even if Russia manages to make significant repairs to the (damaged) bridges, they will remain a key vulnerability," the British said.

On Saturday, the deputy director of the Russian-controlled Kakhovka hydropower plant 60 kilometers (37 miles) upriver from the city of Kherson said one of its generating units was out of service after a Ukrainian missile strike. Arseniy Zelenskyy said further strikes could endanger the Zaporizhzhia nuclear plant because its water intakes use the reservoir formed by the Kakhovka plant's dam.

Days after explosions at a Russian air base in Crimea destroyed up to a dozen aircraft, a Ukrainian presidential adviser said Kyiv should make retaking the Black Sea peninsula that Moscow annexed in 2014 one of its goals of the war.

“Russia started a war against Ukraine and the world in 2014, with its brazen seizure of Crimea. It is obvious that this war should end with the liberation of Crimea," Mykhailo Podoylak, the head of Ukrainian President Volodymyr Zelenskyy's office, wrote Saturday on Twitter. "And also with the legal punishment of the initiators of the ‘special military operation’” - the Kremlin's term for its war in Ukraine.

Ukrainian officials have not claimed responsibility for the explosions Tuesday at the Saki air base in Crimea. Russian defense officials have denied any aircraft were damaged — or that any attack even took place — attributing the blasts to on-site munitions that exploded.