Ottobon, you'll find the source of the Yak-3 "wiping the skies of the German aircrafts" is rooted in a Soviet era authored book of the late 1970's. Since western authors had difficulty in verifying sources, those claims became repeated in subsequent books until the Soviet archives were uncensored and made available to researchers.
So most books that discuss the Yak-3 before 1991, claim that the Yak-3 aircraft participated in the slaughter of the Luftwaffe at Kursk, but do not indicate what units operated them or who the "glorious" pilots were. And you'll notice that in many cases, the passages in those books making this claim are almost exactly the same, word for word.
The reality is, the Yak-1M with the VK-105PF2 was being tested nd was known to be at the battle of Kursk (although it is reported as not having swept the sky clear of German aircraft) and this may have caused confusion, since the -1M was the forerunner of the Yak-3.
It was the 91st Fighter Regiment that began service testing of the early Yak-3 in June of 1944 (source: Soviet Combat Aircraft of the 2nd World War by Gordon and Khazanov) and it was a short time afterward that the Yak-3 saw operational service. Long after the battle of Kursk.
A good book to read, is "Yak fighters of the Great Patriotic War period" (ISBN 5-217-01192-0), by A.T. Stepanets published in 1992. It might be noted that Stepanets worked at Yakovlev, director of flight at the design bureau.
So unfortunately, as exciting as it may sound, there were actually no Yak-3 aircraft at the battle of Kursk.