Helicopter instructor reported a NMAC while on final approach near the 1;000 ft. runway marker. Just prior to touchdown; a single engine aircraft landed below the helicopter creating the near miss.

Narrative

Me and a student were flying on right traffic to Runway 5 (Runway in use). I myself did the right base call and a final call after clearing and turning on to final Runway 5.I remember; that there was quite some traffic on the CTAF frequency at the time and I vividly remember hearing an aircraft calling a final after we were already on short final (I believe we were at end of runway).On short final (being over the runway) approaching the 1000 ft. markers steep at about 40 ft.; we noticed Aircraft Y landed right below us on the runway. I called out the aircraft right away about what's going on and he replied; 'I called out final.'If we were lower (which we were supposed to be if the student was on track); a collision would have been unavoidable. The aircraft was so close and so low; that (ramp personnel said) our downwash from the rotors affected his landing significantly just before touchdown.Communication and situational awareness are in my opinion definitely a factor in this incident. It was a busy traffic pattern. At least 4 planes and us as the helicopter. We; as helicopters; are requested to fly right traffic opposite than the airplanes. Traffic joins together on final where situational awareness must be increased even more by everyone.We had a plane land in front of us and I might have assumed; that this was the last aircraft that I remembered to have heard on the radio calling out a final. I do not want to say; that the other student did not do a call he might have; I just don't recall; neither does my student.We had several factors negatively affecting effective communication. Me instructing while in busy traffic pattern and language barriers of the student in the airplane.Management is now trying to implement new procedures for the airport to where we would stay away of the runway and just use one of the two parallel taxiways for landing.