Kyiv also said Russian forces had launched four drones at Ukraine overnight and that air defence systems had downed two over Kharkiv.
The interior ministry said the missile attack late Wednesday targeted a village near Kupiansk, a frontline town Russian forces have been trying to capture.
"The facade of the two-story building of the medical institution, windows and roof were damaged," it said in a statement.
It added that four people were injured and that a total of 38 people were evacuated.
Russia's defence ministry meanwhile said air defences shot down 11 Ukrainian drones over Russian territory overnight.
"This night, an attempt by the Kyiv regime to commit a terrorist attack with aerial drones against sites on Russian territory was foiled," the ministry said in a statement.
"Air defences intercepted and destroyed drones over the regions of Belgorod (four drones) and Kursk (one drone)", it said.
Around 0330 GMT, four more drones were shot down over the Belgorod region and two over the Voronezh region, it said.
Ukraine has been under persistent aerial bombardment by Russian forces for nearly two years and has appealed to allies in the West to bolster its defensive systems. Thousands of people are estimated to have died in the bombings.
In recent months, Ukraine has upped drone attacks against Russian territory, mainly targeting border regions, but also the capital Moscow and the northern city of Saint Petersburg.
The deadliest such attack occurred in the city of Belgorod on December 30, when 25 people were killed, some 100 wounded and several hundred others evacuated.
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