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This paper studies restricted modules of gap-$p$ Virasoro algebra $\L$ andtheir intrinsic connection to twisted modules of certain vertex algebras. Wefirst establish an equivalence between the category of restricted $\L$-modulesof level $\el$ and the category of twisted modules of vertex algebra$V_{\mathcal{N}_{p}}(\el,0)$, where $\mathcal{N}_{p}$ is a new Lie algebra,$\el:=(\ell_{0},0,\cdots,0)\in\C^{\halfp+1}$,  $\ell_{0}\in\C$ is the action of the Virasoro center. Then we focus on theconstruction and classification of simple restricted $\L$-modules of level$\el$. More explicitly, we give a uniform construction of simple restricted$\L$-modules as induced modules. We present several equivalentcharacterizations of simple restricted $\L$-modules, as locally nilpotent(equivalently, locally finite) modules with respect to certain positive part of$\L$. Moreover, simple restricted $\L$-modules of level $\el$ are classified.They are either highest weight modules or simple induced modules. At the end,we exhibit several concrete examples of simple restricted $\L$-modules of level$\el$ (including Whittaker modules).

Restricted modules for gap-$p$ Virasoro algebra and twisted modules for certain vertex algebras