C172R pilot reported a near mid-air collision with a Cessna that departed in front of them while on short final approach. Pilot went around and established lateral offset from the other aircraft; re-entered the pattern and landed normally.

Narrative

We were conducting a long-distance cross-country IFR training sortie. I was the student pilot for the flight. We entered the RNAV XX at ZZZZZ for ZZZ. ATC released us for VFR into the airport. My instructor was in the right seat. Winds were reported as 90 crosswind. We made distance calls on CTAF after reaching ZZZZZ1. Traffic in the pattern was using runway XY and making calls. We had positive 2-way contact on CTAF with an aircraft in the pattern where we announced our intention to cancel the approach at an appropriate distance out and join the downwind. At 1300 feet MSL; we aborted the approach and I was allowed to remove the foggles. We maneuvered to enter the downwind. We called midfield at 1;200 feet MSL. Abeam the numbers; I noted two aircraft waiting in the runup area. I configured for the final descent and landing. Upon turning base; I called on CTAF; 'ZZZ traffic; Aircraft X base runway XY; full stop; ZZZ.' I rolled out at the completion of the base turn; verified no traffic was coming in from my right; then rolled into final. Upon completion; I rolled out on runway heading; slightly offset to the right. Before I could call 'final'; another aircraft called they were departing followed by them immediately taking runway XY. It took me and my instructor a second or two to process the scene before us before I executed a go-around. My instructor radioed on CTAF that we were going missed and side-stepping right. We moved approximately 200 feet to the right of the runway and began our climb back to pattern altitude. During the climb out; we were about 200 feet to the right and 20 feet aft of the Aircraft Y. We stayed in this position until we attained pattern altitude; where we leveled off as the other aircraft continued to climb and then turned to the left. We waited for a suitable distance between aircraft before turning crosswind for our second; and successful attempt at landing.