The student pilot reported that during the landing roll of a touch-and-go the airplane was off the centerline and he added right rudder, but he "overcompensated". The flight instructor reported that the airplane drifted to the right and was about to exit the runway, so he pushed in the throttle, applied left rudder, and pulled back on the yoke to initiate a climb. The student pilot reported that as the airplane became airborne they heard a "slight bang" and thought the landing gear had struck a runway light. The flight instructor further reported that they conducted a full-stop landing to examine the landing gear, but there was no damage noted. During a postflight inspection at their home airport, it was revealed that the left horizontal stabilizer had sustained substantial damage. The flight instructor reported that there were no preaccident mechanical failures or malfunctions with the airframe or engine that would have precluded normal operation.