With a Muzzle velocity of 2650fps your 110gr bullet will have about 850 FPE at 300 yards. Is that enough to kill a deer? Maybe, depends on bullet design and construction but more important, shot placement. A 22 will kill a deer with a perfectly placed shot.
That brings up the big question, how accurate are you and how accurate is your Mini? The Mini can be moderately accurate, about 2 inches at 100 yards is fairly common. That equates to about 6 inches at 300 yards. Are you comfortable shooting a deer at 300 yards with a 6 inch pattern? Personally I would keep it under 200 yards.
I have a Mini-30, ballistically the 7.62 x 39 and the 6.8 are very similar, I like to stay under 150 yards for MN Whitetails with mine. For longer distances, I use a bigger gun.