Let X be a projective manifold and let $D=\sum_{i=1}^{m}a_iD_i\in |mL|$ be an effective divisor on $X$ with SNC support. Let $f:X\to Y$ be a surjective morphism over a projective manifold $Y$. Write $D=D_v+D_h$, where $D_h$ is the sum of irreducible components of $D$ which are all dominant onto Y. Assume that there exists a Zariski open set $U\subset Y$, so that $X$ is smooth over $U$, $f(D_v)\cap U=\varnothing$, and $D_h$ is relative normal crossing over $U$, i.e. any stratum $\{D_{i_1}\cap D_{i_2}\cap\ldots\cap D_{i_n}\}_{D_{i_j}\subset D_h}$ is smooth over $U$. Let $p:Z\to X$ be a cyclic cover of $X$ obtained by taking the $m$-th roots along $D$, and let $\hat{Z}$ be the normalization of $Z$.
My question is: is there exists a "natural" desingularization $\mu:W\to \hat{Z}$, so that the inverse image $(p\circ\mu)^{-1}(D)$ has SNC support, and $f\circ p\circ\mu:W\to Y$ is smooth over $U$?
Since the singularities of $\hat{Z}$ is locally trivial over $U$, and the singularity obtained by cyclic cover of SNC divisor is at most quotient, I suspect that one might be able to apply some "algorithm" of resolution of quotient singularities so that it will induced a simultaneous resolution of singularities over $U$.
