Starting with a monoidal category with duals $C$, one may consider the category $End(C)$ of endofunctors of $C$. A Hopf monad on $C$ is a bimonad on $C$ with (a generalised notion of the) antipode. Under suitable assumptions for the Hopf monad, this Hopf monad applied on the unit object of the category $C$ is a Hopf algebra in the centre of $C$.
The question is if the notion of Poisson monad in $End(C)$ has been defined somewhere? In the case that such a notion exists, does it return a Poisson algebra object in the category?