  We are extending results from \cite{B-Hurwitz} by building a parallel theory
of simple Hurwitz numbers for the reflection groups $G(m,1,n)$. We also study
analogs of the cut-and-join operators. An algebraic description as well as a
description in terms of ramified covering of Hurwitz numbers is provided. An
explicit formula for them in terms of Schur polynomials are provided. In
addition the generating function of $G(m,1,n)$-Hurwitz numbers is shown to give
rise to $m$ independent variables $\tau$-function of the KP hierarchy. Finally
we provide an ELSV-formula type for these new Hurwitz numbers.
