If one parameter is not enough, you always can add more:

Test m a b = Test { f :: m IO a -> IO b }

This way, if

run :: m IO a -> IO a

then

Test run :: Test m a a

But for other type for your run function

run' :: m IO a -> IO b

you get

Test run' :: Test m a b

So you can have different types in input and output. Anyway, your type 'm' is applied to other two types (m IO a), so it cannot be a monad, because monads have arity 1 as type constructors, i.e. monads have kind (* -> *). Is perhaps 'm' some kind of monad transformer?

Well, that's all I can say from your explanation of the problem! Hope it helps!

Daniel Díaz.