Ukraine has announced that its troops have regained control of territory on the Russian border near the country's second-largest city of Kharkiv, which has been under constant fire since Moscow's invasion began.

The defence ministry said in a statement on social media late Sunday that the Ukrainian troops of the 127th brigade in the Kharkiv region "drove out the Russians and claimed the state border".

In a video released with the statement, a clutch of Ukrainian soldiers in camouflage and holding weapons are gathered around a yellow-and-blue-painted border demarkation post and address President Volodymyr Zelensky.

The military has "reached the border with the Russian Federation -- with the occupying country -- we've done it; we're here," one of the soldiers says.

Ukraine presidential adviser Oleksiy Arestovich on Sunday told local television that Russian troops were being redeployed towards the Donbas region south of Kharkiv after withdrawing, following a Ukrainian counteroffensive.

Since failing to take the capital at the beginning of the invasion in late February, control of Donbas has become one of Moscow's primary objectives -- but Western intelligence has predicted its campaign will stall amid heavy losses and fierce resistance.

"We thank everyone who, risking their lives, liberates Ukraine from Russian invaders," Kharkiv regional governor Oleg Synegubov said in a statement on social media.

"We still have a lot of work ahead of us," he added.

The Ukraine presidency said in a briefing Monday that Russia had continued strikes on the Lugansk region south of Kharkiv.

"Two people were killed and nine were wounded, including a child, as a result of shelling on a Severodonetsk hospital," it said, referring to a major urban hub.

In the western city of Lviv, city authorities said official infrastructure had been hit on May 13 by a cyberattack and that the Ukrainian SBU security services were working to limit the fallout.