Terekhov said there had been three strikes on a city district and that a high-rise apartment block and private homes had been damaged.
Regional Governor Oleh Syniehubov said Russian forces had launched missiles at the city, a frequent target of attacks about 30km from the Russian border.
In neighbouring Dnipropetrovsk region, Governor Oleksandr Hanzha said Russian forces had launched more than 50 drone and artillery attacks.
A drone strike in the city of Pavlohrad killed one person and wounded 10, he said.
Three of the injured were seriously hurt.
Because Kharkiv lies close to the front line, it is repeatedly attacked by the Russian military, both with artillery and with drones.
Massive Ukrainian drone attacks on Russia's Voronezh region killed at least three people and left many more injured, local authorities said on Monday.
Governor Aleksandr Gusev wrote on Telegram that a train had also been hit.
In total there were 39 injured, most of them on the train.
Earlier, two dead and 17 injured had been reported.
Gusev said the train was travelling through a suburb of Voronezh when debris from a downed drone fell on it.
The Russian railway company RZD said one carriage of the train, which was travelling from Anapa on the Black Sea to Moscow, was damaged.
Rail traffic in the region was temporarily suspended because of the attacks.
Russian media reported that the train carried footballers from Lokomotiv Moscow returning from an away match.
A total of 113 drones were shot down in Voronezh and the surrounding region, according to Gusev.
The governor said a business centre caught fire during the attacks on Voronezh.
He also reported damage to industrial facilities, a residential building and a school sports facility, adding that facades and windows were damaged in almost every part of the city.
with DPA
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