It has to be Dremeled out, then filled with a color matched methacrylate, scraped to flush, and polished.

From your description, it is clearly time to have a professional refinish that piece. Stun may be minimized or removed but the most likely solution is the suggestion of Joseph above. Cost effective is subjective but I feel a worthy consideration to refinish.

Thank you. What is involved in refinishing marble counters? Is that something that can be done in place? Is it a huge project and does it restore them like new? If it's something I can only do once or twice, maybe I should wait until the counter is more banged up. Thank you!