A couple of things. Your marble is man made as opposed to marble ... it is made by big chunks of marble + resin or cements to glue them together. You might find how to polish marble but the advice may not apply to man made marble.
Do
not use any acids. Usually the complete surface of marble is polished (using abrasive pastes etc) and the shine - depending on the marble - comes from very find grinding. Man made material may have polish applied to it. I don't have any experience (or bottle rather) to suggest how you might try to polish off the scratches ... but whatever you do use the finest grinding material (steel wool, emery paper or something else). The problem is you will be touching an area wider than the actual scratch and may make it much worse.
I have a tiny problem (similar to yours) and I bought this ...
http://www.briwax.co.uk/products/marblewax.html ... but I have not done anything. Enough dust on the marble covers everything.
If the wood surround is not fitted ... you can experiment with the area that will be covered by the surround.