Flight Instructor reported; while instructing a student pilot on landing; the student had a loss of control and landed hard causing damage to helicopter.

Narrative

Private Pilot Add-On Student enrolled on Day 0 who had Fixed Wing ATP Rating and 15 hours previous in R-22; 2 months prior. I scheduled student with an instructor to do first flight and who would keep the student through need Part 61 Training to meet requirements for add on rating. I instructed the instructor to make the student aware he would have to fly as many dual hours as necessary for him to feel comfortable endorsing him for solo. I was out of office that day; and the instructor messaged me saying that after one more flight the student would be ready for endorsement.The student scheduled to come in on Day 5; for a dual flight and potential solo flight in the same day. I sent the student the Pre-Solo Test to complete prior to coming. However; I lapsed and did not inform the instructor or the student he would need to complete a pre-solo stage check prior to soloing per company policy (being a Part 61 Add-on; I became focused on trying to complete the student as efficiently as possible). The owner/chief pilot corrected me and upon landing from their first flight I made the instructor and the student aware of the needed requirements. I asked them both if they wanted; I was available; and we could complete the pre-solo stage check today. They both agreed; I gave the student the list of required maneuvers and told him I would be ready shorty.Day 5 ~XA:00 First flight conductedDay 5 ~XC:30 Second flight takeoffI completed an electronic weight and balance for our weight and 30 gal of fuel for our aircraft. We were within limits. I checked the forecast; and no weather would have an impact on the flight. Winds were steady; not gusty. Upon arriving at the hangar; I briefed the student on my expectations; CRM; positive exchange of controls; and the general plan for the flight. We conducted preflight; I checked basic system knowledge and items to be more attentive to when conducting the preflight. Student's knowledge was proficient; minor items I had to inform him of (ie. tail rotor drive shaft damper check). Prior to entering the cockpit; we discussed departure and arrival procedures and loss of tail rotor effectiveness; including the importance of always achieving a headwind on takeoff. Again; discussed decision making was proficient.Start up: Student did not use a checklist. Had to be told to roll on the throttle when priming and roll back off. During engine run up had to be instructed to raise collective to 15 MP before conducting Magneto checks. Upon takeoff checks forgot the boost pump; I corrected him. Tower communication and departure safe and to standards. (Checklist usage noted to debrief) Enroute and Maneuvers: About 2 miles southwest of ZZZ; I asked student the emergency procedures for the low fuel light. He responded with return to the airport. I corrected him with POH procedures of landing immediately; I then instructed him on other emergency requirements. We discussed the local area hazards with flight instructor commons and the glider operations at ZZZ1. We continued west to a common public land area where we often train helicopter students about 10 - 15 miles west of ZZZ. We completed a normal landing; then I had him hover taxi for 100 yards; then completed two rapid decelerations. After that we set up to complete a run on landing; we conducted two patterns to run on landings. Student preformed maneuvers to PPL standards and were safe for solo flight. (Emergency items noted to debrief)Autorotations: Finally; I needed to evaluate autorotations from 500' AGL. I briefed that he would enter the auto himself but needed to give me a '3;2;1' countdown prior to. He agreed. The first auto he entered and was going long from the spot picked out; he stated this; passing 100' AGL he began to bring up collective and shortly after initiated a small flare and then quickly nose over to level skids; at this point I did guard the cyclic to keep it forward; so we continued with forward momentum because the rpm droop withhis early collective pull. We were still about 50' AGL I told him to increase throttle which he did; and I verified to allow the engine to fully spool up so we would stop our descent. As we continued forward with stable flight conditions we continued to takeoff. I debriefed him on the auto and then started a discussion on how collective input effects rpms. He had the correlation mixed and kept saying throttle which is typical of fixed wing pilot. I corrected this; told him to continue downwind to discuss how to use collective/rpm management with airspeeds to land at desired location (corrections for being too steep or too shallow/short of the spot). At that point we turned inbound; safe landing zone was assured; landing checks complete. The student entered; 500-100' AGL normal; RPMs in the green; airspeed 50-60 knots.100-80' AGL student initiated up collective75-60' the student began a small flare (enough to bleed airspeed but not build RPMs)50' I rolled on throttle and began pushing the nose forward.30-20' skids level; throttle rolled on but continued RPM droop (winds 15 - 20 knots; no forward momentum achieved)Hard landing: Myself and the student were okay but upon looking behind I saw the tail significantly lower than it should be; all gauges indicated normal but more vibrations than normal. I shut down the helicopter; we stayed in the cockpit until the rotor blades stopped turning. Upon exiting saw the damage. Immediately called the owner/chief pilot and other company dispatchers/administrative personnel.I checked with the student to ensure that they were mentally and physically okay. He agreed and departed the area to meet prior engagements.Of note/learning points: Slow down. Yes; the client was trying to be efficient; but no matter the previous experience I must set the standard to stick to guidelines and standards. I set the student and primary instructor up for failure in not identifying a needed pre-solo stage check prior to due to an unusual circumstance that led to a lapse in judgement. On all flights I guard the controls; I always try and give student room to 'feel' the aircraft and learn from their mistakes; but if safety is ever a concern; I take controls. On stage checks I mentally note to try and give students more room for mistakes (meaning I am guarding controls; but not taking controls as readily in a normal training flight) because if they're soloing or going to an FAA Practical; I need to be sure they can correct their mistakes; again if safety is ever a concern; I take over controls. This incident is not the student's fault; as the instructor I needed to recover sooner. I was a half second too slow on getting the nose over/engaging the throttle and by that point the rpms had depleted enough; winds prevented forward momentum; and the rate of descent was still too high to avoid a hard landing. In past stage checks I have often flown once or twice with the student so I know their tendencies and can give the student a little more freedom on controls but still be able to safely recover in any situation. I had never flown with this student and gave him that freedom which is my fault. There would be no harm in me initiating a go around upon his up collective and demonstrating the standard steps for a safe autorotation to power recovery; then allowing him another attempt/inform him he would need more flights prior to solo endorsement. I am scheduled with the Chief Pilot for a review/standardization flight(s) prior to flying with students again. The designated instructor and I were current on all FAA requirements at this time and within company policy of a standardization flight within the year.