  We consider a class of endomorphisms which contains a set of piecewise
partially hyperbolic skew-products with a non-uniformly expanding base map. The
aimed transformation preserves a foliation which is almost everywhere uniformly
contracted with possible discontinuity sets, which are parallel to the
contracting direction. We prove that the associated transfer operator, acting
on suitable anisotropic normed spaces, has a spectral gap (on which we have
quantitative estimation) and the disintegration of the unique invariant
physical measure, along the stable leaves, is $\zeta$-H\"older. We use this
fact to obtain exponential decay of correlations on the set of $\zeta$-H\"older
functions.
