Pick loose splinters of wood out of the gouge by hand, or sharpen a wood dowel and use the pointed end to pick loose pieces out of the gouge.
Lay a tarp or blanket on the floor. Flip the table over to reveal the unfinished underside of the tabletop.
Wrap 100-grit sandpaper around a sanding block and sand the unfinished side to create enough sawdust to fill the gouge. Sweep the mahogany sawdust onto a plastic plate with a small hand-held broom.
Turn the mahogany table upright or on its side to gain direct access to the gouge.
Lay strips of blue painter's masking tape or other low-tack masking tape around the perimeter of the gouge to protect the undamaged area during repairs. Place the masking tape as close as possible to the edges of the gouge.
Mix the mahogany sawdust with wood glue to form a paste. Pick the paste up with a popsicle stick and press it firmly into the gouge. Add more sawdust paste to the gouge until the mahogany sawdust and wood glue rise just above the surface of the table. Let the wood glue dry overnight.
Wrap 180-grit sandpaper around a sanding block and sand the surface of the repair. Wipe the table with a tack rag to remove excess dust. Wrap 240-grit sandpaper around the sanding block and sand the repair again. Wipe the table with the tack rag to pick up the dust. Wrap 400-grit sandpaper around the sanding block and sand the repair a third time. Wipe the table with the tack rag and remove the tape.
Dip a lint-free cloth into a matching-color, liquid wood scratch cover if the mahogany table has a wood stain finish. Rub a light coat of the scratch cover over the repair and allow it to dry for 20 to 30 minutes. Add more scratch cover as necessary to darken the color and match the table. You can also apply a coat of colored paste furniture wax, let it dry and buff off the excess wax. If the mahogany color is natural, dip #0000 steel wool into paste furniture wax and rub it into the repair, let it dry and buff the area with a soft rag.