Ukraine said Russian forces fired on areas just across the river from the plant, while Russia claimed Ukrainian shells hit a building where nuclear fuel is stored.
Russian rocket and artillery strikes had hit areas across the Dnieper River from Zaporizhzhia Nuclear Power Plant, Ukrainian officials said Sunday, as fears persisted that fighting in the vicinity could damage the plant and cause a radiation leak.
Authorities have been distributing iodine tablets to residents who live near the southeastern Ukraine plant over the past week in case of radiation exposure, which can cause health problems.
Much of the concern centres on the cooling systems for the plant's nuclear reactors.
The systems require power to run, and the plant was temporarily knocked offline Thursday because of what officials said was fire damage to a transmission line.
A cooling system failure could cause a nuclear meltdown.
Russian forces occupied the nuclear plant complex early in the six-month-old war, but local Ukrainian workers have kept it running.
The Ukrainian and Russian governments have repeatedly accused the other of shelling the complex and nearby areas, raising fears of a possible catastrophe.
Heavy firing overnight into Sunday left parts of Nikopol without electricity, said Valentyn Reznichenko, governor of the Dnipropetrovsk region.
Rocket strikes damaged about a dozen residences in Marhanets, according to Yevhen Yevtushenko, the administration head for the district that includes the city of about 45,000.
The city of Zaporizhzhia, about 40 kilometers (25 miles) upriver from the nuclear plant, also came under fire during the night, wounding two people, city council member Anatoliy Kurtev said.
In eastern Ukraine, where Russian and separatist forces are trying to take control, shelling hit the large and strategically significant cities of Kramatorsk and Sloviansk, but no casualties were reported, said Pavlo Kyrylenko, governor of the Donetsk region.
Much of the Donetsk region is held by Russian and separatist forces. It is one of two Ukrainian regions that Russia has recognized as sovereign states.
Periodic shelling has damaged the power station's infrastructure, Ukraine’s nuclear power operator, Energoatom, said Saturday. "There are risks of hydrogen leakage and sputtering of radioactive substances, and the fire hazard is high,” it said.
The governor of Ukraine’s Dnipropetrovsk region, Valentyn Reznichenko, said Saturday that Russian Grad missiles and artillery shells hit the cities of Nikopol and Marhanets, each located 10 kilometres (6 miles) across the Dnieper River from the plant.
But Russian Defense Ministry spokesman Igor Konashenkov said Ukrainian forces had fired on the plant from Marhanets.
Over the past day, 17 Ukrainian shells hit the plant, with four striking the roof of a building that stores nuclear fuel, he said.
Associated Press reported it was not immediately possible to verify either account.
The United Nations' atomic energy agency has tried to work out an agreement to send a team in to inspect and help secure the plant. Officials said preparations for the visit were underway, but it remained unclear when it might take place.
Europe's biggest nuclear plant was disconnected from the national grid for the first time in its history because of fire damage. ITV News' Ian Woods reports.
Ukrainian President Volodymyr Zelenskyy said it was essential for International Atomic Energy Agency representatives to get to the plant as soon as possible and to help keep it "under permanent Ukrainian control.”
“The situation remains precarious and dangerous,” Mr Zelenskyy said Friday in his nightly address.
“Any repetition of (Thursday's) events, i.e., any disconnection of the station from the grid or any actions by Russia that could trigger the shut down of the reactors, will once again put the station one step away from disaster.”
Ukraine has claimed Russia is using the power plant as a shield by storing weapons there and launching attacks from around it.
Moscow, for its part, accuses Ukraine of firing on the nuclear complex.
The dispute over the plant led Russia late Friday to block agreement on the final document of the four-week-long review of the UN treaty that is considered the cornerstone of nuclear disarmament.
The draft document of the Nuclear Nonproliferation Treaty review conference criticised Russia's takeover of the Zaporizhzhia plant.
The deputy head of Russia's delegation said the conference became “a political hostage” to countries that were trying “to settle scores with Russia by raising issues that are not directly related to the treaty.”
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Elsewhere in Ukraine, one person was killed and another wounded in Russian firing in the Mykolaiv region, local government officials said. Mykolaiv city is an important Black Sea port and shipbuilding center.
The governor of the eastern Donetsk region, Pavlo Kyrylenko, said Saturday that two people were killed in Russian firing on the city of Bakhmut, a significant target for Russian and separatist forces seeking to take control of the parts of the region they do not already hold.
The UK government said Saturday that it was giving Ukraine underwater drones and training sailors to use them to clear mines from the ravaged country's coastline.
Mines laid in the Black Sea during the war have hampered seaborne exports of Ukrainian grain to world markets, although an agreement reached in July has allowed shipments to resume along a single corridor.
More than 1 million metric tons of Ukrainian foodstuffs have been shipped since the start of August under the Black Sea grain deal, the UN said.
The flow of grain under deal has driven down prices, reduced the risk of food insecurity and allowed the World Food Program to restart wheat purchases from Ukraine for drought-hit countries such as Ethiopia and Yemen.