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We extend the Dong-Mason theorem on the irreducibility of modules fororbifold 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 operatoralgebra, $g$ 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 inequivalentirreducible modules. We prove that $W$ is an irreducible weak $V^{\left\langleg\right\rangle }$-module. This result can be applied on irreducible modules ofcertain 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 certainapplications in the cases of the Heisenberg and Weyl vertex operator algebras.

On irreducibility of modules of Whittaker type for cyclic orbifold vertex algebra