  The purpose of this paper is to show how one can extend some results on
disorder relevance obtained for the random pinning model with i.i.d disorder to
the model with finite range correlated disorder. In a previous work, the
annealed critical curve of the latter model was computed, and equality of
quenched and annealed critical points, as well as exponents, was proved under
some conditions on the return exponent of the interarrival times. Here we
complete this work by looking at the disorder relevant regime, where annealed
and quenched critical points differ. All these results show that the Harris
criterion, which was proved to be correct in the i.i.d case, remains valid in
our setup. We strongly use Markov renewal constructions that were introduced in
the solving of the annealed model.
