  This paper focuses on directed polymers pinned at a disordered and correlated
interface. We assume that the disorder sequence is a q-order moving average and
show that the critical curve of the annealed model can be expressed in terms of
the Perron-Frobenius eigenvalue of an explicit transfer matrix, which
generalizes the annealed bound of the critical curve for i.i.d. disorder. We
provide explicit values of the annealed critical curve for $q=1$, 2 and a weak
disorder asymptotic in the general case. Following the renewal theory approach
of pinning, the processes arising in the study of the annealed model are
particular Markov renewal processes. We consider the intersection of two
replicas of this process to prove a result of disorder irrelevance (i.e.
quenched and annealed critical curves as well as exponents coincide) via the
method of second moment.
