During a touch and go at Mid State Airport, upon touch down and lowering of the airplane's nose on runway 24, the student pilot and CFI noticed a snow bank across the runway at the intersection of runway 16/34. The CFI instructed the student to takeoff immediately. The student applied full takeoff power, and just after lift off they heard a 'thump'. The airplane's left main gear impacted the snow bank and was torn away from the airplane causing substantial damage to the horizontal stabilizer and left wing and flap. The student and CFI continued the takeoff and returned to their base airport where a fly by was accomplished and a damage assessment was accomplished by ground personnel. The CFI made an emergency landing and the airplane occupants exited the airplane with no reported injuries. The CFI stated that they did fly over Mid State Airport prior to landing and that they did not see anything on the runway. The manager of Mid State Airport reported that the person who snow plowed runway 16/34 was new to the airport and was not aware that the other runway was in use, hence, did not remove the plowed snow from the intersecting runway. There were no Notices to Airmen published concerning a closure of runway 24.