  We extend the Dong-Mason theorem on the irreducibility of modules for
orbifold vertex algebras from [C. Dong, G. Mason, Duke Math. J. 86 (1997)]
305-321] for the category of weak modules. Let $V$ be a vertex operator
algebra, $g$ an automorphism of order $p$. Let $W$ be an irreducible weak
$V$--module such that $W,W\circ g,\dots,W\circ g^{p-1}$ are inequivalent
irreducible modules. We prove that $W$ is an irreducible weak $V^{\left\langle
g\right\rangle }$-module. This result can be applied on irreducible modules of
certain Lie algebra $\mathfrak L$ such that $W,W\circ g,\dots,W\circ g^{p-1}$
are Whittaker modules having different Whittaker functions. We present certain
applications in the cases of the Heisenberg and Weyl vertex operator algebras.
