  This paper deals with cyclic covers of a large family of rational normal
surfaces that can also be described as quotients of a product, where the
factors are cyclic covers of algebraic curves. We use a generalization of
Esnault-Viehweg method to show that the action of the monodromy on the first
Betti group of the covering (and its Hodge structure) splits as a direct sum of
the same data for some specific cyclic covers over $\mathbb{P}^1$. This has
applications to the study of L\^e-Yomdin surface singularities, in particular
to the action of the monodromy on the Mixed Hodge Structure, as well as to
isotrivial fibered surfaces.
