  We study the random pinning model, in the case of a Gaussian environment
presenting power-law decaying correlations, of exponent decay a>0. We comment
on the annealed (i.e. averaged over disorder) model, which is far from being
trivial, and we discuss the influence of disorder on the critical properties of
the system. We show that the annealed critical exponent \nu^{ann} is the same
as the homogeneous one \nu^{pur}, provided that correlations are decaying fast
enough (a>2). If correlations are summable (a>1), we also show that the
disordered phase transition is at least of order 2, showing disorder relevance
if \nu^{pur}<2. If correlations are not summable (a<1), we show that the phase
transition disappears.
