Disclaimer: Your mileage may vary, but this is how I'd tackle it:
I've had luck filling chips like that with "Quickwood" or similar material. Quickwood is almost the exact same color as the tan of the wood around the ring, so once the Quickwood dries, you might not have to color match the tan of the wood colored circle. I always leave the Quickwood chip filler a little lower than the surrounding playfield so I have a slightly recessed area to fill with paint and clear.
Once the Quickwood dried, I'd raid my stash of acrylic paints and touch up the black and purple areas, and tan if needed. I'd probably have to custom blend the purple and tan (wood tone). Once done with the colors, I would expect that the area would be slightly recessed (if I planned correctly) so I could flow clear into it. I have acrylic clear that I'd probably used, but in a pinch, I've had good luck with water thin super glue. You just don't want to get that super glue anywhere else on your playfield, machine, hands, floor, etc., LOL.
Once finished, the top of the clear over the touch-up should be at playfield level, and the repair will basically be imperceptible if done well. Even if not done well, it will probably escape notice, especially from the player's perspective.