GA Flight Instructor reported a near miss at PSM airport with another aircraft on final while landing on a training flight. ATC instructed the other aircraft to go-around and offset from the runway; the Instructor observed the aircraft in close proximity; performed a takeoff and turned away from the aircraft immediately then returned for a landing.

Narrative

The two aircraft involved were a Small aircraft (aircraft X; which I was the flight instructor onboard) and a Small Aircraft (aircraft Y); both being used for flight training. Aircraft X was cleared for a touch and go on runway 34 from the left downwind by the tower controller at the PSM airport. Aircraft two was told to report aircraft X in sight; which upon doing so was cleared for the option on runway 34 as aircraft Y. As aircraft X was touching down; the tower controlled advised aircraft Y to complete a go around and to offset to the left side of the runway. Inside of aircraft X; an audible 'traffic' alert was heard and the MFD displayed aircraft Y '+0' feet above. Upon glancing to my 7 O'clock; I observed aircraft Y continuing a descent towards the runway; unclear of if they were conducting a go around or not. I confirmed the flight controls from my student; applied full power; and conducted a prompt go around as I was unsure of if aircraft Y was going to continue to the runway and land on top of me. As I began a climb; aircraft Y began a climb and I began a right hand turn to offset to the right side of the runway; between the altitude of 50-200' AGL. At this point; the two aircraft were side-by-side on opposite sides of the runway at the same altitudes. Without being prompted; I advised the tower controlled that I had traffic in sight; and the controlled cleared me to begin a right crosswind turn at my discretion; which I had fully committed into at approximately 300 feet AGL. The tower controller asked aircraft Y if they had aircraft X on the approach for a second time and aircraft Y admitted that they did not have aircraft X in sight at the end; hence the lack of adequate spacing between the two aircraft. Aircraft X then made a full stop landing.