Cessna 172 Flight Instructor reported an unstabilized approach during a training flight resulting in the aircraft touching down short of the runway. The instructor took control; initiated a go around and safely departed to another airport.

Narrative

Conducting pattern work at ZZZ fourth lap in the pattern we were cleared for the option after flying on the extended downwind for Runway XX. We turned base adding flaps 10 around between 85 - 95 kts. Then turned on to about a 3.5 mile final for Runway XX as we turned out we were above glide path; and the student corrected and flew the papi's one white three red down. Throughout final we added flaps 20 and 30 aiming for the numbers and on short final the sink rate became excessive we then encountered a loss of 5 - 10kts occurred short final which increase the already exessive decent rate. The student was told to add power to arrest decent rate; but power was not added quickly enough; instructor took controls and initiated a go around before ground contact; the plane touched down roughly 5ft short of Runway XX threshold in the grass and then climbed out and departed back to ZZZ1. Probably cause: LLWS as well as too low on glide path short final. Suggestions for mitigations: As the instructor; I should have initiated a go-around as soon as the initial sink rate was determined to be excessive and been firmer with my student on flying papi's (two while two red) the entirely of the way down instead of letting them go slightly below glide path (one white; three red). Also aiming point should have not been Runway XX numbers considering conditions and should have been moved further down the runway to the touchdown zone. This alone resides on me the instructor and not on the student.