Vadym Filashkin, governor of Donetsk region, on Wednesday said 11 people had been injured in the daytime Russian attack on residential buildings in Kramatorsk.
The governor of southeastern Dnipropetrovsk Region, Oleksandr Hanzha, said there had been three Russian strikes near the region's largest city, Dnipro, injuring eight people and triggering a large fire. Three people were in hospital in serious condition.
President Volodymyr Zelenskiy, speaking in his nightly video address, said Russian forces had struck food storage areas and a postal depot, deploying drones and missiles.
Photographs posted online showed several buildings ablaze and billowing smoke engulfing the surrounding areas. More than 100 firefighters had been deployed.
In the southern city of Kherson, an overnight drone attack destroyed 36 apartments in a residential building. Regional Governor Oleksandr Prokudin said one person had died in the incident.
In Russia's border region of Bryansk, Acting Regional Governor Yegor Kovalchuk said a Ukrainian drone strike had killed a crane operator working for the local utility.
And the Moscow-installed governor of parts of Ukraine's eastern Donetsk region under Russian control, Denis Pushilin, said the death toll in a drone attack on a bus early on Wednesday had risen to eight, with 12 injured.
Reuters could not independently verify the different accounts.
Meanwhile, Ukraine's stepped-up strikes deep inside Russia have enabled Kyiv to negotiate the end of the war on an equal footing, Zelenskiy said, just after one of the barrages struck an oil terminal and naval base hundreds of kilometres away.
For months, Kyiv's troops have been conducting attacks on Russian fossil fuel industry sites, at times daily, aiming to cut Moscow's main funding source for the war and pressure the Kremlin for talks.
"Thank God that today we have security guarantees that allow us to end this war on equal footing with the Russians in any format of diplomacy," Zelenskiy told reporters in Kyiv, speaking alongside NATO Secretary General Mark Rutte.
Zelenskiy said it was "only a question of time" for Ukraine to increase the scale of the strikes, which forced some of the Russian refineries to suspend operations and boosted morale among Ukrainians, who have lived under constant threat of Russian drones and missiles for more than four years.
As Ukraine has ramped up long-range attacks on Russia, Kyiv's troops on the battlefield appear on their best footing in years, analysts say.
Russia's spring offensive is sputtering, partly a result of Ukrainian counterattacks that have kept its gains minuscule, they said.
Zelenskiy said he was ready to meet Russian President Vladimir Putin, which Kyiv has long insisted was the only way to resolve the main stumbling block in so far stalled talks -- the issue of eastern Donbas. Russia failed to fully occupy the area during its full-scale invasion and has demanded Ukraine pull back troops in the region that it still controls.
"I am ready for direct talks with Putin to bring this war to an end, rather than waiting for when all will resolve every conflict in the world before our turn finally comes," Zelenskiy said, in an apparent reference to the US-brokered talks and current US focus on its war on Iran.
