We have been here before and not that long ago though that time with a burn in the checkering, it would be a good idea to look up the posts. There are two methods that will do a reasonably hidden repair. The first is a method used for hiding cigarette burns in furniture in the past. you will need to clean the burn down to un effected wood using abrasive paper, and producing slopes edges into the burn removing viable straight edges if you do not your eye will see them instantly detracting from the repair. Next colour the new looking wood to match the rest of the stocks surface. from here you can go two ways traditional this would involve you melting clear shellac to fill the depression just above the surface then reducing the filling level with the surface using fine abrasive paper. when all is level you turn the shellac clear by wiping over its surface quickly with a a lint free cloth soaked in alcohol. The second method would be to use a hard setting clear cast Epoxy sanded down to the surface though you will have to use a dedicated plastic polish to bring back the optical clarity to the repair.
Another method would be to use a piece of wood veneer cut from a piece of walnut with a similar grain run, you must clean up the burn leaving clean sharp edges so that your veneer patch fits in tightly when glued into place. The hard part is to adjust the patch colour to blend in with the rest of the stock. I did go over the methods used in detail in the previous postings.
