The most famous is club Drive at Rudika sq. Parties are held on Fridays, Saturdays and all national holidays (New Year, Christmas, 8th of March etc.) there.
Another club is called Cactus. It is situated somewhere in the area called Alexeyevka.
There’s also a gay-friendly cafe right in the center of Kharkov (near the entrance of metro “Universitet”). It is called Versay, and there can be found some gays and lesbians in the evening.