Stuart,

Average coming up after this. First have hardware removed from sink. Second, lightly hone (200 grit) area. This will remove any etching and open the stone. Third, mix a poultice to the consistency of mashed potatoes. Not too thick,  not too runny. Mix into poultice some Iron out. White powder form purchased at Home Depot. Add a cup to cup n a half. Mix well. Apply your poultice to the stains. Cover thoroughly and at a thickness of at least a 1/4 ". Now cover your poultice with plastic. Pat down to remove all air and tape edges. Leave your poultice on the stain (covered) for 24-48 hours. Next, remove plastic and allow mud to dry. During drying your poultice is pulling out the stains.
When poultice is cracked and dry, remove and clean area. Repolish as needed.

The reason I suggest iron out on this stain is it looks like it could be from hard water. Iron likley present. Go get em!

Ron