  In this paper, we study Virasoro vertex algebras and affine vertex algebras
over a general field of characteristic $p>2$. More specifically, we study
certain quotients of the universal Virasoro and affine vertex algebras by
ideals related to the $p$-centers of the Virasoro algebra and affine Lie
algebras. Among the main results, we classify their irreducible
$\mathbb{N}$-graded modules by explicitly determining their Zhu algebras and
show that these vertex algebras have only finitely many irreducible
$\mathbb{N}$-graded modules and they are $C_2$-cofinite.
