The city, located in Ukraine's northeast just 30 kilometres (18 miles) from Russia's border, has faced intensifying aerial attacks in recent months.
Moscow also launched a major assault on the wider Kharkiv region last week, capturing several border villages and triggering Ukraine to evacuate thousands from the area.
"This pressure is aimed at pushing people to leave, to worry them, this is the tactic of the Russian Federation," Kharkiv Mayor Igor Terekhov told AFP on the scene of one of the strikes on the city.
A spate of aerial attacks on residential areas wounded 20 people, including three children, the regional prosecutor's office said in a post on Telegram.
Officials were inspecting the aftermath of one attack on a multistorey building, next to a children's toy shop.
Teddy bears were still lined up on the destroyed shelves of the shop that workers had already begun cleaning.
An AFP reporter saw residents walking around their homes, hugging each other and stepping on broken glass and metal.
"A guided aviation bomb blew up and virtually tore apart half of the (tenth) floor, destroying the nearby apartments on the higher and lower floors," Sergiy Bolvinov, the head of the region's Police Investigation Department, told AFP.
Ksenia Gorvits, 32, was playing a synthetiser at home when the strike "smashed" into her place, she said.
"Everything was like in slow motion, like in some film. It took me a while to realise what had happened," she said, standing outside the building.
"Miraculously, nothing got on me."
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