  Making use of the weighted Mourre theory developed in [GJ1], we show the
limiting absorption principle for Schr{\"o}dinger operators with perturbed
oscillating potential on appropriate energy intervals. We focus on a certain
class of oscillating potentials (larger than the one in [GJ2]) that was already
studied in [BD, MU, ReT1, ReT2]. We allow long-range and short-range components
and local singularities in the perturbation. We improve known results, the main
novelty being the presence of a long-range perturbation. A subclass of the
considered potentials actually cannot be treated by the usual Mourre commutator
method. Inspired by [FH], we also show, in some cases, the absence of positive
eigenvalues for our Schr{\"o}dinger operators.
