Oleh Syniehubov says a baby and two children - aged four and seven - were among the victims.
Photos from the city near the Russian border show a number of houses on fire after a petrol station was hit, reports BBC.
One man was reported injured in Ukraine's southern city of Odesa in a separate Russian drone attack.
Visibly emotional, Kharkiv Mayor Ihor Terekhov said "the Russian aggressor is ruining our city".
"But we will overcome. We will win," he said in a video message from the scene of the Russian attack.
He added that 14 houses were set ablaze.
Russia's military has made no public comments on the reported strikes.
Russian President Vladimir Putin launched a full-scale invasion against Ukraine in February 2022.