M20K Student pilot reported his CFI lost control of the aircraft during an attempted go-around following the reporter's bounced landing.

Narrative

My CFI and I were practicing takeoff and landings within the pattern. This was a part of transitioning to a newly acquired aircraft. Prior to the incident; I performed a takeoff and a subsequent landing with my CFI present. We taxied back to the active runway; as directed by ATC Ground. Once cleared by Tower we departed; the takeoff was uneventful and we remained in the pattern. On downwind GUMPS check was performed; the aircraft was configured for landing. On final; a short-final check was performed and I once again confirmed the appropriate configuration for landing. The approach speed was approximately 75 kts. and overall descent rate was also appropriate; verbally confirmed by my CFI. I distinctly remember hearing my CFI state; 'looking good...' The aircraft descended over the numbers and the flare was appropriate; however when the aircraft touched down; it bounced. My CFI announced; 'my controls!'; I confirmed; 'your controls!' and controls were immediately released to my CFI; he applied full throttle in an attempt at a go-around. The aircraft deviated to the left and soon we were no longer on the runway; but in the grassy median separating the runway from the taxiway. I believe at this point my CFI cut the power and was able to avoid other aircraft on the taxiway. Our aircraft crossed over the taxiway; continuing the leftward trajectory; and came to a full stop on the grass median; left of the taxiway.Tower called us and asked if we needed assistance; my CFI replied that we indeed needed assistance. They subsequently asked if we needed medical assistance; I responded that medical is not necessary; after confirming that my CFI and I were not injured. I switched the fuel to off; electrical system was turned off and my CFI and I exited the aircraft. The nose gear was partially collapsed; the propeller was damaged; however control surfaces; wings; main landing gear all remained intact. It seems that the left turning tendencies were underestimated; especially during a go-around procedure.