The flight instructor reported that the pilot receiving instruction was attempting to perform a soft field landing on runway 31 during the second landing of the flight. The flight instructor reported that the wind was from 160 degrees at 12 knots, gusting to 18 knots. During the landing flare, the airplane was “getting pushed” off centerline. The flight instructor informed the pilot receiving instruction to put in full power and do a go-around. The airplane’s nose wheel struck a light generator on the side of the runway causing the airplane to skid across the taxiway and into the grass, which resulted in substantial damage to the wing. The flight instructor reported that there was no malfunction or system failure of the airplane before the accident.