There's a flap which controls whether fresh air comes into the cabin from the outside or is recirculated within the cabin.
From your description & video it could be the flap, for some reason, isn't fully opening or closing.
This could be due to a foreign object preventing the flap from reaching the fully open/closed position and consequently the motor is continually trying to move the flap and the gear mechanism is jumping teeth - hence the clicking.
Or, there are broken/missing teeth on the gear mechanism, causing the clicking, which again could mean the flap isn't reaching the full open/closed position and consequently the motor is constantly operating.
Disclaimer: I'm not a vehicle mechanic so could be wrong.
(Heating/Ventilation units in modern cars are usually buried deep behind the dashboard and a PITA to get to)
Yes, I am a miserable bugger. But that's your problem, not mine.
Anything I post may be complete b****ks. Especially after whisky consumption. Don't say I didn't warn you.