The white piece actually is screwed in with a metal washer and a plastic washer. The part that has those wobbly extensions snap onto the motor. Then the washer with the screw go in. Must be aligned correctly for the remote to position the mirror.
Before you try buying a new piece, have you tried reinstalling this piece since it doesn't look damaged from the picture. If you do need to replace it then I would be looking for the entire side view mirror. I know it is more expensive. But, you would be happier if the mirror doesn't fall out if that piece was damaged.
Note: Replacement mirrors don't always snap in easily. I just replaced mine with heated mirrors. The mirror just wouldn't push on. Had to open the clips on the mirror for it to snap on.
Edit: Just looked at picture again. The center of the piece is broken. There will be no way to screw that piece in. You will need a new piece. Still believe a whole new side view mirror would be best. Don't know what other damage the mirror might have absorbed. Good luck.