Apply a moderate layer of wax to steel wool by rubbing hard paste wax with the steel wool. For deep scratches, use 360-grit sandpaper instead of steel wool.
Rub the steel wool over the scratches in the wood veneer in the direction of the grain. Rub lightly and just enough to work the paste wax in and minimize the scratches in the furniture. Do not sand through the wood veneer layer.
Buff the paste wax with a chamois until the scratches and paste wax disappear and a shiny gloss is left behind.