The Russians were never in Kharkiv, unless you mean Kharkivska Oblast, in which case, no they haven't left.
The latest from the Russian side is reported fighting at Buhruvatka, which clashes somewhat with reports that the UAF had reached the Russian border, though they may have made up a victory in the classical sense by claiming to take a village that was never in Russian control. The situation is very fluid and what is true one day, however, is untrue in the other. The Ukrainians are attempting to bait the Russians into weakening the attack towards the Donbas from the Izyum direction but the Russians are well aware of this, and there is no indication of any such action being taken. A river crossing from the Ukrainians is also a possibility, but has not happened yet. The Russians are holding in defense of Lyptsy and Kozacha Lopan after retreating without a fight from many other villages in the face of a UAF counteroffensive, and there were rumors a few days ago that the UAF was planning a large offensive towards there, but nothing has of yet materialized.
"Giving up" the Donbas front is neither a necessity nor a possibility. The at most 10,000 men involved in this counter offensive (according to Russian forces, many TDF, though this is unconfirmed) aren't nearly enough to force Russia into such a thing.