This is likely a common "bibb" style valve, where the water comes through a valve seat. The water is stopped by a bibb washer on the end of a moving stem; the washer seals by moving up against the valve seat.
The picture below shows a common bibb-style valve (yours would not be identical but same general parts and principle):
A clip or clamp on the hose will do nothing to keeping the hose "up in" a cavity. A pressure fitting is needed, that expands into the cavity. Check with the plumbing expert where you purchased the device.
Use an appropriate length braided stainless steel covering hose. If you must use adapters, stainless is best.
No bi-metal interactions that way. Else ...
What you are calling a white pipe is likely the body of the mixing valve. The installation as it is now looks carefully done: a nice circular hole, and a schluter kerdi seal around the hole (the orange plastic ring inside the hole). I wouldn't jump to the conclusion that it sticks out to much. You probably just bought the wrong brand of handle.
You're best ...
If you cannot cut the pipe back I would make a spacer. I would make it a couple of inches larger than the cover plate the same shape (if round make the spacer round) by making the spacer larger and the same shape it will look like it was planned for decorative reason instead of a mistake. If you need multiple layers make each one larger by a similar amount ...
You should try to loosen/remove the piece still in there.
it should just unscrew. The cover you removed doesn't appear to be part of the original drain. If you can remove the remaining piece, you can get a whole new piece to screw back in and no water will be pooling around.