In a statement on social media, Synegubov described the Russian attack on Bohodukhiv as “brutal”.
He said that the family that was in the hit house had recently evacuated from a town near the Russian border, trying to escape constant shelling.
“It was their first night in the new place,” the regional head added.
Bohodukhiv, north-west of Ukraine’s second largest city of Kharkiv, has been under almost daily Russian attack in recent days.
On Monday, a 41-year-old woman and a 10-year-old boy were killed in the town, local officials said.
Russian forces also dropped seven bombs on the city of Slovyansk on Tuesday, Donetsk regional head Vadym Filashkin said on Telegram. Three people, including an 11-year-old girl, were killed.
Moscow has increased pressure on the area by targeting transport and energy infrastructure with a record number of ballistic missiles, Ukrainian President Volodymyr Zelensky has said.
A state of emergency has been declared over the energy situation in the Kharkiv region by local authorities.
It comes after a so-called “energy truce”, which was meant to allow Ukraine’s energy sector a reprieve during -20C (-4F) temperatures, expired last week.
Tens of thousands of people across Ukraine are experiencing severe power cuts, and many households have no running water or heating.
Zelensky has said the US wants the war with Russia to end by June, and that both sides had been invited to the US for talks next week.
Meanwhile, a massive attack by Ukrainian drones on Volzhsky, in the Volgograd region in Russia, was reported on Wednesday night.
Regional Governor Andrei Bocharov said an apartment was damaged in the strike and a fire broke out at a factory.
Russia launched a full-scale invasion of Ukraine in February 2022.