First, this is the right site. Second, it's a bit difficult to see the complete problem due to the light reflection, so if you could take a shot looking straight down at the surface, that would help. That being said, here are some suggestions/thoughts:
1) While I can't be positive, I believe that the finish color is a combination of:
A) some stain/dye color applied to the wood,
B) then a bit of the same or different color is mixed with some diluted finish, and sprayed on the stained surface,
C) then a couple coats of clear finish applied.
The reason this is important is that it's probably going to be difficult to have the damaged areas match the rest of the surface. So to get the best repair you're probably going to have to strip the surface
2) What you can do is determine the type of finish. Take a cotton swab dipped in lacquer thinner and rub it on an inconspicuous spot. If the finish comes off or gets sticky it's probably a lacquer type finish. If not it's a poly or industrial multi-part finish.
3)There is a method that can be done in place of stripping the entire surface. It may just reduce the damage, but it's not a guarantee. Since you'd have to strip the surface anyway, if this doesn't work, no harm has been done. If it does work, you're ahead of the game. Use this If - and ONLY if, it's a lacquer finish. To use this method you can try the following.
The basic concept is to make a small ball of cotton (an old, worn t-shirt works well). Place this cotton ball it inside another, larger,12" x 12" piece of cotton & wrap the big piece around the small ball making a ball out of this. Wrap the excess from the large ball around the index finger. Add some lacquer thinner to this ball, or "pad" (this is called "charging the pad") and pat it against a flat surface. You want the pad to be damp, not wet. Then use a pendulum type motion to have the pad touch the damaged surface & come right off...almost like a plane coming in for a landing and then taking off again. DO NOT LET THE PAD STOP ON THE SURFACE. If the pad starts to stick, then charge it (add more lacquer thinner) again. The trick is to dampen the cloth just enough so it leaves the appearance of a comet’s tail as you wipe. (You can practice by wiping across a flat board). If you get the cloth too wet, the lac thinner can soften the finish too much and dull the sheen or smear the finish.
Let us know how you make out
Bob "Boardman" Borders