The solo student, pilot reported that, during the landing flare, the gear-retracted warning horn sounded. The propeller struck the ground and he applied full throttle and initiated a go-around. The engine began "bucking," so he decided to lower the landing gear and land on the remaining runway. He applied forward pressure to the yoke to level the airplane, but the airplane touched down hard and came to rest upright, off the left side of the runway. The student recommended on the NTSB form 6120.1 that he should have verified the configuration of the landing gear prior to landing. He added that he did not use a pre-landing checklist. The airplane sustained substantial damage to the right wing. The student reported that there were no preaccident mechanical failures or malfunctions with the airplane that would have precluded normal operation.