How do I seal a leaking plughole fitting?
This basin trim piece you are referring to is usually connected to the tail piece flange with a very fine thread. If has been in place for a long time, they can be difficult to remove. You will need to insert something like needle nose pliers in the top and hold the tail piece from the bottom and unscrew it. Clean out all the old plumbers putty and replace it by making a "rope" of plumbers putty in your hand, place it around the top edge of the hole in the basin, then reinstall the trim piece into the tail piece section. Tighten it so it pushes out all the excess putty, and is tight.  Clean up the excess, return it to your can. Job done.