Russia has sporadically targeted Odesa, a Black Sea port, and a week ago, Ukraine said at least eight people were killed in a strike on the city.
"As a result of a missile attack in the Odesa region, the runway at Odesa airport was damaged. Its further use is impossible," the Ukrainian military said.
There was no immediate word on the strike from the Russian military.
Russian forces have also pounded Ukraine's eastern Donbas region over night. In the town of Dobropillia in Donetsk, the shockwave from a strike blew in the windows of an apartment building and left a large crater in the yard.
One resident, who gave only his first name of Andriy, said his partner was in a room facing the yard at the time of the attack and was knocked unconscious.
"Thank God the four children were in the kitchen," he said, standing in the destroyed living room.
Residents sifted through their belongings to see what could be salvaged.
"At around 9.20am this happiness flew to our house", said Oleh, who also gave only his first name, speaking with sarcasm. "Everything is destroyed."
Moscow hopes to take full control of the eastern Donbas region, made up of Luhansk and Donetsk, parts of which were already controlled by Russian-speaking separatists before the invasion.
The General Staff of Ukraine's Armed Forces said in a daily update that the Russians were trying to capture the areas of Lyman in Donetsk and Sievierodonetsk and Popasna in Luhansk, adding they are "not succeeding".
The war has turned cities to rubble, killed thousands and forced 5 million Ukrainians to flee abroad. Russia turned its focus to the south and east after initially surrounding the capital, Kyiv.
Moscow calls its actions a "special operation" to disarm Ukraine and rid it of extreme anti-Russian neo-Nazism fomented by the West. Ukraine and the West say Russia launched an unprovoked war of aggression.