Flight Instructor with student reported an NMAC in the traffic pattern.

Narrative

Student and I were doing short approach touch and goes at ZZZ on [Runway] XXL. On my aircraft ADS-B the plane is identified as Aircraft Y. Aircraft Y was ahead of us in the pattern and was doing (as far as I could tell) normal landings. We got cleared for a short approach being notified we will be 'inside the archer' (meaning Aircraft Y) This happened about XA55Z. Up until this point the student was making most all of the calls; student was not fast enough to make response call so I responded instead. When I responded Tower repeated the clearance; hence there is garbled sound if you listen to the archived frequency tapes. Tower heard the end of my transmission and said 'thanks; make a short approach; short approach'. Again my student did not transmit response immediately so I responded while it appears Tower repeated. We stepped on each other on the frequency. Some short time later Tower told Aircraft Y to go around and to offset to the right. Aircraft Y did not respond so tower repeated. Aircraft Y responded to the call though the tapes sound unintelligible so I am not sure if Aircraft Y knew to offset. Tower told us that Aircraft Y is going around to the right side. We responded and acknowledged. Aircraft Y told Tower it was a near mid air collision. Tower told Aircraft Y again to offset to the right. During this entire time I was looking outside for this aircraft and only saw them on the go around. When we had them in sight we maneuvered as necessary to avoid them and ended up touching down further down the runway. It is hard to tell how close/far they were. After landing I got a call from the flight school because Tower called them wanting to speak to me. I called Tower and they only asked for my name and phone number and how close they were. I did not receive a pilot deviation or any accusatory remarks during the call. They simply said they needed information to fill out a report. I believe this could have been avoided if Tower did not clear us for a short approach and told us to do a normal landing. It could have also been avoided if the other plane had gone around sooner. It could have also been avoided if I made the decision to not continue with the landing and instead extend downwind and follow the Archer. I had thought the Archer was further away than what it actually was.