Non-weight modules over gap-$p$ Virasoro algebras

In this paper, we study non-weight modules over gap-$p$ Virasoro algebras, including Whittaker modules, $\mathcal{U}(\mathbb{C} L_0)$-free modules and their tensor products. We establish necessary and sufficient conditions for universal Whittaker modules to be irreducible and study the structure of irreducible Whittaker modules. The $\mathcal{U}(\mathbb{C} L_0)$-free modules of rank 1 are classified and the irreducibility of such modules are determined. Moreover, the irreducibility of tensor products of $\mathcal{U}(\mathbb{C} L_0)$-free modules of rank 1 and irreducible restricted modules is also determined.