PA-28 Flight Instructor with student reported engine roughness and power loss while executing a missed approach. Flight crew returned to land at departure airport.

Narrative

On the Day 0; I planned to do a cross-country flight from ZZZ1 to ZZZ2 and practice instrument approaches at ZZZ with my student. The cross-country went perfectly; so we returned and practiced RNAV XX at ZZZ as planned. Inbound for RNAV XX; the Tower cleared us for a low approach. Upon reaching minimum; we applied full power; achieved a positive rate; and put the flaps up normally. However; upon intersecting Runway X and Runway XX during the low approach; we both felt engine vibration and roughness. The Instructor grabbed the control; but we did not have enough runway to pitch down and land on [Runway] XX so we turn right for Runway XY to put it on the grass since that seems like the only option at the time. Suddenly; the Instructor tried to adjust the throttle again; and the engine came back. However; we began losing power again; and the RPM dropped to around 1;700. We [requested priority handling] at that point; and the Tower cleared us to land on any available runway. We successfully landed on Runway XZ.