In recent weeks Mykolaiv has been hit almost daily as Ukrainian troops seek to push into the neighbouring Kherson region. Seven people were killed Friday in an attack near a bus stop.
The Ukrainian presidency said its forces had carried out strikes on Russian military warehouses and positions behind the lines in Kherson region.
In Ukraine's second city of Kharkiv, rockets from an S-300 surface-to-air system destroyed part of an educational facility in a strike in the early hours of Saturday, local authorities said.
Firemen extinguished a blaze caused by the fire and there were no reports of casualties, the authorities said.
Kharkiv, close to the Russian border in northeast Ukraine, has also been subjected to frequent bombardment by Moscow's forces stationed nearby.
The governor of the eastern Donetsk region, where Moscow is focusing the brunt of its offensive, said that six civilians were killed and 15 wounded by strikes on Friday.
A bus station in the city of Sloviansk, a key target for the Kremlin's forces, was damaged in an attack Saturday morning but no casualties were reported, he said.