In an emergency, the handbrake is very unlikely to help. It is likely to put you in a slip. Especially in a car with ABS and traction control, where normal brakes are far more effective.

The only emergency it is useful for is when you really have to pull a 180 right now and tight, as the automated systems do not expect this situation and will relax the brakes while you're deliberately trying to spin. I wouldn't trust even an above average drive with pulling that off - it's not being practiced. Usually it would be a desperation move when there's no space and a head on collision is likely.

it‘s a handbrake, not an emergency brake.

I'm feeling conflicted because my driving instructor said that you could and maybe should use handbrake for additional barking power in an emergency, but car's manual says that it shouldn't be used and may be damaged if used while the car is moving..

Do not use the handbrake for emergencies. You don't have antilock brakes with the handbrake and it will probably throw you into an uncontrollable skid rather than slow you down.

How would it be damaged if used while the car is moving? It's just a cable that activates the rear brake caliper independently of the hydraulics.