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.
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