A day like today in 1943, Kharkov was definitely liberated by the Red Army after two years been captured by both sides. The city and its nearest region was a very active war zone during those first two years of the Soviet-German War. If we look at the German terminology, there were four battles for the city, named First Battle of Kharkov, Second, etc.
The last one (named Fourth Battle of Kharkov by the Germans and Belgorod-Kharkov Strategic Offensive Operation by the Soviets) finished on August 23rd, 1943. The city was captured the last day of Operation Polkovodets Rumyantsev, one of the Soviets’ operations carried out after Operation Citadelle. The Operation was performed by the Voronezh and Steppe Fronts and was commanded by Ivan Konev. In addition to the capture of the city, the German 4th Panzer Army and Army Detachment Kempf were seriously damaged.
Besides the capture of Kharkov, on August 23rd the Battle for Kursk came to its end. It started on July 5th and after one month and a half, the Soviet Union regained control of several cities along with several thousand square kilometres. The Red Army suffered incredible casualties but the German Armed Forced would never recover from this defeat.