Can a thick piece of glass be polished to remove the scratches?
Start by applying a thin layer of paste wax. Let it dry, then buff with a clean cotton cloth. If the scratches are shallow enough, that might be all you need. The wax fills the divots and helps light reflect as if the surface is smooth.
If that doesn’t do it, but the scratches are still so shallow that you can’t feel them with a fingernail, you can probably have them buffed out with a special polishing compound.
If you want to attempt the repair yourself, check out products sold for rubbing away scratches in automobile glass, such as the Diamondite Glass Repair Kit. It sells for $50 to $60, depending on whether you get the system for hand application or the one for machine application. The machine kit includes a polishing pad that fits into an electric drill.
If your table has deeper scratches, attempts to rub them away with a buffing compound might leave the surface looking worse than it is now.