STUDENT PLT LOST CONTROL OF SR20 DURING GAR. INSTRUCTOR PLT TOOK CONTROL OF THE ACFT; BUT LANDED OFF THE RWY.

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MY STUDENT AND I WERE ON FINAL FOR RWY 18 INTO ZZZ. THE WINDS WERE 190 DEGS AT 18 KTS. THE APCH SPD WAS 90 KTS. WE WERE INITIATING THE FLARE AT ABOUT 85 KTS. MY STUDENT INITIATED THE FLARE TOO HIGH AND THE RESULT WAS A HARD LNDG FOLLOWED BY A BOUNCE. I IMMEDIATELY INSTRUCTED MY STUDENT TO GO AROUND. DUE TO THE FACT THAT MY STUDENT HAS SHOWN SEVERAL SAFE GAR'S BEFORE; I DID NOT TAKE CTL. HOWEVER; MY STUDENT FAILED TO GIVE SUFFICIENT R RUDDER. UPON INSTRUCTING THE STUDENT TO ADD THE R RUDDER; EXCESSIVE R AILERON WAS ADDED INSTEAD. THE R AILERON AND R RUDDER COMBINED PUT THE ACFT INTO A BANK OF ABOUT 20 DEGS. THE STUDENT THEN TOOK OUT THE PWR AT 50 FT AGL. I TOOK CTL AND ROLLED THE WINGS LEVEL AND ATTEMPTED TO RETURN THE PWR TO FULL. BY THE TIME I HAD THE ACFT UNDER CTL; THERE WAS LITTLE I COULD DO BUT MAINTAIN WINGS LEVEL. WE TOUCHED DOWN ABOUT 30 FT TO THE SIDE OF THE RWY. FAILURE TO INITIATE A GAR WHEN NOT ESTABLISHED ON THE FINAL APCH AT THE PROPER AIRSPD CONTRIBUTED TO THIS INCIDENT. IMPROPER AILERON AND EXCESSIVE RUDDER INPUT IS WHAT CAUSED THIS INCIDENT.