Name of the town (6,000 today) is Shatsk = Шацьк in Ukrainian. During Interwar Period it was the battleground between the Poles and the Red Army in 1939. In that time frame it was in Luboml district. Today it is a district center w/appx 13 villages. It is possible that extended??? family ended up in Kharkiv. Stalin was not very happy w/anyone who voluntarily or involuntarily landed up in Germany. Their families landed up in resorts like Sakhalin in Siberia and Khazakstan and in the mines of the Donbas in Ukraine. Many people have returned to their homelands. I will check for the surname in both Kharkiv and Shatzk.
Maybe Grandmother was from Shatzk and Grandfather was from Kharkiv??? Until you get some paperwork from Bad Arolsen, everything is still a guess.
I did check for Hukow in Shatzk and found none.