Ccorazone

Remove old veneer careful not to damage underlying wood. With sort of rough sandpaper ( 80 or 100) remove as much of the old glue as possible making sure you don't change the shape in any way. Turn upside down, trace the outline of the shape on paper but make it a tiny bit larger. Use the same species of wood to replace what you took off. Veneer sheets can be purchased online.
Use the paper outline to cut new shape. Test an edge of the veneer to see if it can be cut with scissors without shattering the wood. If it does shatter then that method is out. Next method is an extra sharp exacto blade. Lay a straight edge(ruler)to guide your cut. Make a series of cuts. Do not attempt to cut all the way through in one cut it will shatter the edge of the wood. Glue in position on lamp place waxed paper or plastic wrap on that then place something heavy on top to weigh to down ( several heavy books) as the veneer as it dries. Make sure the veneer does not shift position.
Once the glue is completely dry wrap fine sandpaper around a block of wood. Sand until the edges are even, stain to match the rest of the lamp add clear topcoat put it back together and enjoy..I may have left a small step but I think you get the idea. Hope this helps.