The pilot reported that during the engine run-up prior to the first flight of the day, he noted there was no resistance on the control knob when he checked the operation of the carburetor heat and there was a slightly “sluggish” return in the RPM as it returned to normal. The engine operated normally during this flight. When he applied carburetor heat during the engine run-up for the second flight of the day, he noticed “RPM needle wiggled down 50” then returned to the set RPM. Everything appeared normal during the takeoff until the airplane reached about 200 feet above the runway when it felt a little “soggy.” The pilot stated he raised the landing gear to reduce drag, but it did not seem like the engine was producing power. The pilot then extended the landing gear and decided to land the airplane in a field off the end of the runway. The landing gear, which was not locked down, collapsed during the landing and the left wing was substantially damaged when it contacted a runway end identifier light. Post accident inspection of the airplane and engine revealed the carburetor heat bracket between the valve shaft and the cable was fractured off and missing. The carburetor heat valve was in the open position.