What is the benefits of controlling the Asian carp?

There are tons. Depends on which kind, there are a few.

The black carp feeds on native mussels and snails, some of which can be already endangered.

Grass carp can alter the food webs of a new environment by altering the communities of plants, invertebrates and fish.

Silver carp feed on the plankton necessary for larval fish and native mussels.

Hope this is what you were looking for.