If the scratches aren't super deep, you could use a headlight restoration kit, that has various grades of abrasives to clear up fogged and lightly scratched acrylic headlight covers. If scratches are deeper, you might need to start with a more aggressive sandpaper, like wet 400 (maybe). Don't over do it.

Use a gradual grit protocol on a random orbital sander. See if a 220 removes the scratches; if not drop down to a 100 or so, back to 220, etc., all the way up to 500 then go to a gray ScotchBrite pad. You'll need some Finesse It on a buffer to get it to clear. You'll work in gradually larger circles until it's blended and gone.

See these guys. They are the best in cleaning and repairing acrylic products.

It depends upon how deep the scratches are as to what method will work. Those look too deep to me to go to liquid polish right away. Can't hurt though. If they don't remove them, go mechanical, then back to the polishes.

You also could email the moving company and include your photos. See what they will do for you. The guys were not professional in their handling of your furniture.