  Let $V\rightarrow X$ be a surjective quasi-projective morphism between
irreducible quasi-projective varieties over a number field, where $X$ is
smooth. We prove that there exists a non-empty Zariski open subset of $X$ such
that the $S$-integral points in it are covered by subpolynomially many
geometrically irreducible subvarieties, each lying in a fiber of the mixed
period mapping. We also prove analogous theorems for semi-simplicial varieties
and abstract variations of mixed Hodge structures. They are based on recent
works by Brunebarbe-Maculan and Ellenberg-Lawrence-Venkatesh. As an
application, we prove that there are subpolynomially many $S$-integral Laurent
polynomials with fixed reflexive Newton polyhedron $\Delta$ and fixed non-zero
principal $\Delta$-determinant. Our results answer a question in
Ellenberg-Lawrence-Venkatesh.
