PRC Tower Controller reported failed to provide separation based on incorrect RADAR and transponder data resulted in evasive action and a NMAC event.

Narrative

Aircraft X called inbound from the NW for a full stop; I told him to enter a right base to a 5 mile final for R (Runway) 21R; which he read back. At the time I had 4 aircraft in my traffic pattern doing touch n go's. The first 3 aircraft had asked for short approaches to the runway; when Aircraft Y was downwind; Aircraft X was turning final; he overshot the final and turned back to join about a 4 mile final. I told Aircraft X to use caution that he overshot final; to which replied 'I didn't overshoot the final I am established now'; at this time his transponder was showing him maneuvering back to final; I told him 'maybe it's my radar but it shows your track overshooting'; he then responded 'well we overshot final a little bit; we are joining final now'. According to his radar target he was on a 4 mile final; I had an aircraft short final and told Aircraft Y that his short approach was approved and that he was number 2 for the runway and cleared him touch and go. I went on and sequenced the other aircraft in my pattern; Aircraft Y started his short approach and then Aircraft X said 'hey tower there is a guy turning base we are on a 2 mile final do you want us to do a 360 for spacing'; I said 'absolutely not; there is traffic in my downwind; that aircraft is making a short approach ahead of you; you are on a 2 mile final that is plenty of space' he immediately came back and said 'no we are on a mile final (at this time the only radar target on final showed was on a 2 mile final; when I clicked on it I realized that it was Aircraft Z who was landing the parallel runway) that is when I realized his transponder had dropped off the radar and he wasn't giving me accurate reporting points. He then said 'we are going around' by the time I saw him on final he was at apprx 060 ft; I told him go around fly straight out; I told him his transponder appeared to have disappeared from the radar to which he said his transponder was working fine. I then turned his crosswind midfield and resequenced him and he landed without incident. Aircraft X called later and said he 'almost had a midair'; I cannot speak to that because the aircraft he was referring to was almost touching down and about 006 below him when I saw his go around.It was a very busy session and it didn't help that I was getting false information from the pilot of Aircraft X and that his transponder had dropped off the radar. We don't use the radar for separation at this facility but with so many student pilots training in our traffic pattern it is a great tool to use to help sequence and locate traffic; also the approach end of runway 21R is more than a mile from the view of the tower.