During a training flight to practice crosswind takeoffs and landings, with gusty wind conditions on runway 32, a gust of wind lifted the left wing of the Cessna 172N during the landing flare. The student pilot inadvertently applied right aileron and the right wing contacted the runway surface, resulting in damage to the right wing and aileron. The certificated flight instructor (CFI) then assumed control, and the airplane drifted right of the runway and gained approximately 10 feet of altitude. The throttle control was bent about 45 degrees by the student during the abnormal runway contact, which prevented the CFI from performing an aborted landing maneuver. The damage to the right aileron also prevented the CFI from returning to the runway surface. The airplane landed in the snow next to the runway and neither pilot was injured. The accident airport did not have weather reporting capability; however, airports with weather observations in the local area at the time of the accident, recorded winds generally from 250 to 290 degrees at 15 knots with a peak gust of 25 knots. The CFI reported that there were no mechanical malfunctions with the airplane prior to the accident.