  For any cocommutative Hopf algebra $H$ and a left $H$-module $V$, we
construct an operad $\mathcal{P}^{cl}_H(V)$, which in the special case when $H$
is the algebra of polynomials in one variable reduces to the classical operad
$\mathcal{P}^{cl}(V)$. Morphisms from the Lie operad to $\mathcal{P}^{cl}(V)$
correspond to Poisson vertex algebra structures on $V$. Likewise, our operad
$\mathcal{P}^{cl}_H(V)$ gives rise to the notion of a Poisson pseudoalgebra;
thus extending the notion of a Lie pseudoalgebra. As a byproduct of our
construction, we introduce two cohomology theories for Poisson pseudoalgebras,
generalizing the variational and classical cohomology of Poisson vertex
algebras.
