  Let $H$ be a cocommutative Hopf algebra. The notion of Lie $H$-pseudoalgebra
is a multivariable generalization of Lie conformal algebras. In this paper, we
study some higher structures related to Lie $H$-pseudoalgebras where we
increase the flexibility of the Jacobi identity. Namely, we first introduce
$L_\infty$ $H$-pseudoalgebras (also called strongly homotopy Lie
$H$-pseudoalgebras) as the homotopy analogue of Lie $H$-pseudoalgebras. We give
several equivalent descriptions of such homotopy algebras and show that some
particular classes of these homotopy algebras are closely related to the
cohomology of Lie $H$-pseudoalgebras and crossed modules of Lie
$H$-pseudoalgebras. Next, we introduce another higher structure, called Lie-$2$
$H$-pseudoalgebras which are the categorification of Lie $H$-pseudoalgebras.
Finally, we show that the category of Lie-$2$ $H$-pseudoalgebras is equivalent
to the category of certain $L_\infty$ $H$-pseudoalgebras.
