After Russia fired four Kalibr missiles on the southern port city of Odesa from a warship in the Black Sea, Ukraine's air force said. The country's western-backed air defences shot down three of them.
But one missile hit a food warehouse, killing three employees and wounding seven, said Oleg Kiper, the head of the region's military administration.
"There may be people under the rubble," he added.
Another six other people were wounded after a business centre, shops and a residential complex in central Odesa were damaged "as a result of air combat and the blast wave", he said.
Overnight, in the eastern cities of Kramatorsk and Kostiantynivka, Russian missile strikes killed three people and destroyed dozens of private houses, regional authorities said.
A blast in Kramatorsk left a huge crater in a street of one- and two-storey family houses, an AFP correspondent said.
The blast completely demolished some houses, and several more around lost their windows and roofs.
The attack killed at least two men, while several other people were wounded and evacuated for treatment.
Police officer Anastasiya Korzun, 33, told AFP she and her husband had escaped their damaged house and then others trying to dig neighbours out of the rubble.
"My colleagues from the police were on the scene in 20 minutes to help," Korzun said, showing cuts on her arm from flying glass.
- Shelling kills six -
Russia said it had launched multiple strikes overnight, targeting Ukraine's troops and "foreign mercenaries" as well as warehouses containing foreign-made weapons.
"All the assigned targets have been hit," the defence ministry said.
Kyiv also reported that Russians had a day earlier shelled a van in northeastern Ukraine, close to the border with Russia, killing six people.
Ukrainian prosecutors said the Soviet-era van was hit near the village of Seredyna-Buda, on the border with Russia in the northeastern region of Sumy. Four of the victims were foresters, they said.
Moscow has intensified its nightly attacks on major Ukrainian cities in recent weeks while Kyiv has launched a long-awaited counter-offensive to reclaim Russian-occupied territory.
Ukraine said Wednesday that its forces had retaken around three square kilometres (about one square mile) of territory and advanced in some areas as far as 1.4 kilometres, in the last three days. Fighting was continuing near recaptured villages.
The latest strikes came as the death toll from Tuesday's missile strikes on Kryvyi Rig -- the hometown of President Volodymyr Zelensky -- rose to 12.
Authorities in the central Dnipropetrovsk region, which includes Kryvyi Rig, also reported a fresh Russian drone attack overnight.
Although Kyiv says it is making gains after launching its counter-offensive, Putin on Tuesday claimed his forces were inflicting "catastrophic" losses on Ukrainian troops.
- IAEA chief to visit plant -
Russian President Vladimir Putin said during a Kremlin meeting with war bloggers that Russian forces were suffering from diminishing stockpiles of some military equipment, pointing in particular to attack drones and missiles.
Kyiv insisted that Ukraine's push, bolstered with Western weapons and training, had had "certain gains, implementing our plans, moving forward".
According to military analysts, Ukraine has not yet committed the bulk of its forces in its counter-offensive, and is currently still testing the front with probing attacks to determine weak points.
In recent days, Kyiv has claimed to have re-captured several villages in the eastern and southern regions.
UN nuclear chief Rafael Grossi was expected at the Moscow-occupied Zaporizhzhia atomic power plant on Wednesday.
But Russian news agencies reported the visit had been delayed by a day.
Kyiv and the International Atomic Energy Agency (IAEA) did not confirm the delay.
The safety of Europe's largest nuclear power plant, located in Ukraine's southeastern region of Zaporizhzhia, has been a concern since Russian forces seized it over a year ago.
Those concerns have been exacerbated by the breach of the Kakhovka dam that provided the cooling water for the plant.
While in Kyiv on Tuesday, Grossi said that while there was "no immediate situation", the water level in the cooling pond was of concern.
The IAEA has warned that the dam disaster, which has left dozens dead and missing, further complicated "an already precarious nuclear safety and security situation" at the plant.
Kyiv has accused Moscow of blowing up the dam on the Dnipro River, while Russia has blamed Ukraine.
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