The student pilot reported that on his first night training flight he was practicing takeoffs and landings. He stated that during his third approach, as he was turning from base to final, the precision approach path indicator (PAPI) displayed four red lights, indicating that he was below the recommended approach path. The flight instructor told him that he was too low and to add power. The student pilot added power and the far left PAPI light started to turn from red to white. He reported that on short final, he reduced engine power to idle and that during touchdown, the airplane collided with a runway threshold light. The airplane sustained substantial damage to horizontal stabilizer spar. The flight instructor reported no preimpact mechanical failures or malfunctions with the airframe or engine that would have precluded normal operation.