Stain on teak table

[AlanS]

Built a teak veneer table years ago.   Took off the tablecloth to find these 'stains'.   I thought that the areas would bounce back with some oil...but no results.   Thought I would gently sand....no positive results.   Aside from trying to sand the entire tabletop (and risk breaking through the thin veneer), what can I do to get the entire top back to 'tip-top' shape.    The stains are probably water based...having gone through the tablecloth