Flight Instructor reported a ground strike while taking control of aircraft from the student who was unable to maintain directional control during landing training. After a safe landing; minor damage to right wingtip and tail eye bolt was reported.

Narrative

I was teaching a lesson doing pattern work. Winds were reported as 240deg at 8 and Runway XX was in use. Another aircraft became disabled on Runway XX; so the ground controller gave us taxi instructions to Runway XY; which we used for the entirety of the lesson. During the lesson; the winds picked up and Tower began to give us the wind information every time we were cleared for a touch-and-go. The wind information varied but was about 270 deg -290 deg gusting to the high teens at the end of the lesson. Runway XX was still closed so we remained on Runway XY. On the final landing; we planned on a full stop and had been cleared to land. The student was flying the plane and I was monitoring. As we landed; the plane veered to the left. I added full right rudder and aileron but our trajectory did not change. As we had only been on the runway momentarily; I decided to go-around instead of going off the side of the runway. I took control; adding full power; setting the flap control to 10deg; and raising the nose to the climb pitch attitude. The aircraft pitched up and rolled to the right. I heard an impact noise as the plane left the ground. I had regained directional control at that point and decided that it would be best to put the plane down on the remaining runway. I landed again and taxied to the ramp under our own power. Upon inspection; there was damage to the right wingtip and the tail eye bolt was bent. My student and I were both fine.