You can use some of the very thin 3M sanding sponges, the green ones. Cut it with a scissors,put in in the slot and rotate with your kids wooden pencil. Pretend you are trying to light a fire like you did in Boy Scouts, rolling the pencil between your palms. That should provide just enough abrasive to sand off the burn marks easily and inexpensively.

Just recut it and as I once learned "dust it a bit" but make sure your setup is right. You may also want to put some dry lubricant on the bit, may help a bit with the burn. If you don't want those options, take a dowel with the right diameter and use adhesive sandpaper and clean it up with that.