I have a little kit for such spots. It's almost the same as "slap some varnish on it" but before doing the slap, you sand the wood and loose varnish with a bit of sandpaper. Then you tape it off and slap on some varnish. If the color of the wood is black, don't slap that varnish on before doing a little bleaching with a bit more care (note, not much more care) so your slapped on varnish doesn't seal in the black/gray color:
1. sand the flaking stuff off. This only takes a couple seconds. Don't whine about it just do it.
2. with an acid brush (the sort of little brush used for flux and soldering) dab a bit of Davis FSC (fiberglass stain remover...see Amazon has it here w/free Prime
shipping http://amzn.to/2t4Jk10 )
3. let that sit a few minutes and then wipe/wash off with
water.
4. let that dry for long enough you can tape it (so you can remember to come back to it later after it is fully dry) with a bit of blue tape.
5. Slap some varnish on the bare spot all the way to your tape. Your tape is going to make it so you don't have a "feather" edge that will
lift before you do the real varnish job.
That's it.