Flight Instructor reported they did not hear or acknowledge ATC cancelling their takeoff clearance resulting in an aircraft on short final initiating a go around to avoid them.

Narrative

We were holding short of Runway XX when Tower cleared another aircraft in the downwind for a short approach. After clearing the other aircraft; Tower cleared us for takeoff on Runway XX ahead of the aircraft on short approach. My student pilot read back the takeoff clearance as I cleared final and opposite end of the runway and checked the MFD (Multifunction Flight Display) for traffic. There wasn't any targets showing on the MFD so I assumed the other aircraft was in the downwind to base phase. As we were on the takeoff roll; the other aircraft declared that they were going around and asked Tower if they had cleared us for takeoff. Tower stated that we were not cleared for takeoff but told us a few minutes later that he did clear us for takeoff but then cancelled our takeoff clearance shortly after. My student and I believe Tower had told us to cancel our takeoff clearance as we were reading back our takeoff clearance so we were stepping on each other and neither we nor Tower heard each other. Tower hadn't followed up to confirm that we had acknowledged the cancelled takeoff clearance. No pilot deviations were issued.