Your engine actually runs with only half a piston ring?!
First hint: <b>Don't</b> run the engine with only a piece of a ring. There's a good chance you'll score your cylinder, or bits of the ring might break off and get wedged between the piston and cylinder or (worse) get into the crankcase and do the equivalent damage to your engine as a wrecking ball hitting a bungalow.
In answer to your question about power in an engine with rings vs. without. You'll have oodles more power <i>with</i> rings. If you test your compression now (with half a ring) I'd bet you are getting somewhere around 50lbs per square inch. A properly running engine should be at 100lbs minimum.
You've got no choice. You have to have at least one ring (preferably two) on your piston.