THE STUDENT & INSTRUCTOR PLT (CFI) WERE PRACTICING LANDINGS. AFTER MAKING 3 TOUCH-&-GO LANDINGS, THE ENG LOST POWER AT APRX 200 FT AGL AS THEY WERE ON THEIR 4TH TAKEOFF. THE CFI REPORTED THAT HE TURNED THE ACFT ABOUT 25 DEG IN AN ATTEMPT TO GLIDE TO A BEAN FIELD & AVOID TALL CORN. HOWEVER, THERE WAS INSUFFICIENT ALT TO AVOID THE CORN. AFTER LANDING IN THE CORN FIELD & ON MUDDY GROUND, THE ACFT NOSED OVER. AN EXAM OF THE ENG & ACFT REVEALED NO PREIMPACT MECHANICAL PROBLEM THAT WOULD HAVE RESULTED IN A LOSS OF POWER. THE ENG WAS STARTED & RUN TO 1500 RPM WITH NO PROBLEMS NOTED, EXCEPT THAT IT COULD NOT BE OPERATED AT A HIGHER RPM DUE TO A BENT PROP. THE TEMP & DEW POINT WERE 78 & 77 DEG, RESPECTIVELY. ACCORDING TO ICING PROBABILITY CHARTS, SERIOUS CARB ICING CONDITION WOULD HAVE EXISTED AT IDLE POWER. HOWEVER,THE CFI STATED THAT THEY HAD USED CARB HEAT ON THE DOWNWIND OF EACH PATTERN, THEN PLACED IT IN COLD FOR EACH TAKEOFF.