Sanding marble will remove any protective coating and allow you to work with the raw stone. For small surfaces use a hand sander; for floors use a floor buffer.
This procedure will allow the work you do to form a consistent finish over the entire area. You can use various sanding methods.
Use light grit diamond abrasives specifically formulated for fine finishing. Marble is fairly soft and aggressive grit may create scratches that are worse than the ones you are trying to remove.
Use progressively finer grits until a mirror-like polish is achieved.
Polishing powders or liquid formulas are another option. Apply powders or polish using a floor buffer or hand sander with a soft cloth pad.
You can use floor finish strippers before applying polish to remove any existing finish that may have become cloudy over time. Polishing powder can also remove water stains and minor scratches.
Simply polishing over damaged finishes may allow the imperfections in the old finish to show through the new.
Another choice for restoring polish to marble to coating it with clear polishing wax or urethane. These coatings have the potential to turn yellow and should only be used on dark floors.
Wax and urethane create a thin seal over the stone and may be used as a protective coating over top of other polishing applications.