Just a single Betta in 10 gallons with filtration should result in a very clean tank, so I'm guessing your problem is algae?

If so, then ramshorn snails would help.

Otherwise, it's cruel to keep a single cory cat without others of its own kind, as they are schooling fish, so no fewer than 6 of each species in any tank.

Also, cory cats need their own food and would only add to the load in your tank.

Ghost shrimp might easily become snacks for a voracious predator like a Betta, even if he ate them in bits. He does have sharp teeth, even if they are too small for you to see them.

He would also eat baby ramshorn snails, and keep their numbers in check.

I put 2 red ramhorn snails in my Betta's tank, and after a year, there are still only 2 adult snails, a fat and happy Betta and- no algae.