Flight crew reported that on training flight in the base turn to destination airport the student accidentally bumped the throttle to a higher power setting and while attempting to correct the power setting inadvertently moved the mixture lever lower resulting in loss of left engine power. Instructor took control of the aircraft and made a single engine landing and safely exited the runway.

Narrative

During a multi-engine training cross-country flight; my student and I were on approach to landing at ZZZ Airport. We had been given instructions to enter a left base for Runway X; which we were in progress of complying with when our instructions were changed to us needing to enter a left downwind for Runway X when we were already within three miles of the field. There was other traffic in the area; including GA single-engine traffic; commercial airline traffic; and private jet traffic. Given this relatively busy moment; I had my eyes outside the aircraft scanning for traffic while my student configured the aircraft for landing. When we were abeam the approach end of X; we were cleared to land. I thought this was somewhat strange because of the traffic that was in the area; but I did not have any traffic in sight visually but could see traffic on ADS-B that I had my student account for this by flying a pattern that would allow for adequate spacing. Somewhere between the base and final turns; my student accidentally bumped the left throttle lever up to a higher power setting while trying to adjust their kneeboard. In the subsequent panic because of the bank and yaw of the aircraft; my student reduced power to the engine using the throttle lever; but also moved the mixture lever lower than it should have been; causing the left engine to lose power. The mixture levers on this model of the aircraft do not have safety locks as most newer aircraft do; meaning that the levers can be moved freely. My student immediately recognized the power loss; but neither of us recognized at the time that the power loss had occurred because of human error; our inattention to this fact likely due to being caught off guard by the control of the aircraft. We both immediately attributed the power loss to mechanical error due to the age of the aircraft. As all of this was happening on final approach; we were in sight of ZZZ Tower. Because of the power loss on the engine at such a low altitude; we did bank and yaw significantly; which Tower likely saw; but I immediately took control of the aircraft and stabilized the flight path for landing. Given the low altitude; I did not try to restart the engine; as it had stopped wind milling because of our approach airspeed and was not a factor in maintaining control of the aircraft. Somewhere between 75-150 ft. AGL; Tower issued us instructions to go-around; likely because of our perceived unstable flight path. I immediately replied 'Unable' and landed the aircraft within the touchdown zone of the runway. Because I was so close to the ground; my primary focus was maintaining control of the aircraft and getting it on the ground safely; therefore; I did not declare an emergency nor inform Tower of our engine out situation; which likely was not visible to them. At this time; no other traffic had been cleared to land behind us; and the one aircraft that was ahead of us had cleared the runway and was in the parking area.Also of note; while we were on the base leg of the approach; I heard Tower instruct personnel in an airport ground vehicle to remove something from one of the taxiways and to inspect that taxiway so that aircraft would be able to clear Runway X. During our approach and landing; I did not hear Tower issue instructions to any other aircraft or ground vehicles to cross or even hold short of Runway X; meaning that the runway environment should have been clear for us to carry out the landing that we had previously been cleared for. If we were cleared for landing on a specific runway; then that means that all taxiways and other paths for egressing or accessing that runway already should have been inspected and cleared for use by aircraft. Never at any point did Tower inform us to be cognizant of crews working in the vicinity of any runways or taxiways.After landing and clearing the runway; I noticed that the left mixture lever was lower than it should have been and attributed this to being the reason for uslosing power when we did. I did not bother trying to restart the engine while on the taxiway because of the amount of time that would have taken given the finicky engine controls and behavior of the particular aircraft we were flying; something we had been aware of prior to and during flight. Upon taxi-in and shutdown; our aircraft was approached by what appeared to be an airport ground supervisor. They questioned us as to why we were unable to comply with ATC's direction for us to go around and mentioned that they saw us taxiing on one engine. I explained the situation as I did here. They left and appeared to make a phone call to someone.Some factors that likely attributed to this event are listed:Fatigue - to carry out this training flight; both my student and I woke up at two o'clock in the morning to make a commercial flight that would get us in the vicinity of our training airport on that day. The commercial flight was followed by a 45-minute drive to the training airport; most of which occurred before sunrise.Unfamiliarity of the aircraft - I am relatively new to the aircraft; but I felt like I had the proficiency to train another multi commercial pilot/CFI to fly the plane; something I also have the experience to do. However; I had not flown the plane in a while; but did go out before the lesson to do some landings and emergency procedures so that I would be ready in the event of a true emergency.Airport congestion - as mentioned; there seemed to be a lot going on at the airport; with both incoming traffic and on the ground. Also as mentioned; if Tower wanted to have one of the taxiways cleared; then they could have told us to extend our downwind leg; or cleared us to land on Runway YY; which we later did use for departure after taking a break and getting our aircraft reconfigured.Inattention to trainee's actions - I was pilot in command for the flight as I am an MEI; and a completely neglected to keep an eye on my student during this critical phase of flight. Also as mentioned; Tower seemed to be relatively indecisive as to what they wanted traffic to do both on the ground and in the air; which caught my attention; warranting me to keep my eyes outside the aircraft for any potential traffic conflicts. This is what led me to direct my attention away from what my student was doing as it was their second landing of the day and I felt confident that they would do a good job. I allowed their error and their response to the error of mistakenly bumping the throttle then moving the mixture go too far. I should have intervened as soon as we got a power increase on the left engine; but I did not.Though I am not wanting to shift responsibility away from myself as an instructor; however; it was relatively workload inducing to receive two back-to-back conflicting instructions from Valdosta Tower. We were first told to enter a left base for Runway X. When just about established on that base; we then got instructions to enter a left downwind; from which we were cleared to land. On a very short final; we received instructions to go-around; which as previously described; we were unable to comply with. That was two sets of conflicting instructions for our aircraft all within ten minutes of each other. We all make mistakes at times; but this was some of the most unacceptable performance that I have seen from a Controller.