Can my High Gloss Black speakers look new again?

Meguiar's "ScratchX". Automotive super fine polish removes swirl marks on $500,000 cars with $20,000 paint jobs.
It WILL work for your finish.
Costs about $10 a tube. at most automotive stores.
i use it on CeeDees and DVDs too.

PS use it with a soft %100 COTTON cloth!
Some claim diapers are the best!
I prefer cotton Tshirts that are 100% cotton but stretchy knit. work excellent

...or a 3M microfiber cloth....and I second that suggestion about the automotive polish...

Another option to the Meguiar's, and also from the automotive world is 3M Swirl Mark Remover (part no. 39009) This is especially formulated for dark color cars, and was recommended by show car polish manufacturer Sal Zaino when I needed to remove some scratches and water marks from my black Corvette. Sal's suggestion carried a lot of weight with me because he is averse to applying ANY abrasive product to automobiles, and for him to recommend 3M Swirl Mark Remover meant it must be effective and gentle. It worked for me. Use a sponge applicator and go very lightly. Of course, we are all assuming your speakers have a paint finish akin to Merlin loudspeakers, and not gloss laminate. If you have gloss laminate, I'd suggest testing any auto polish on an inconspicuous area...like the bottom of the speakers.

Good luck!