  We study the bi-dimensional $q$-Potts model with long-range bond correlated
disorder. Similarly to [C. Chatelain, Phys. Rev. E 89, 032105], we implement a
disorder bimodal distribution by coupling the Potts model to auxiliary
spin-variables, which are correlated with a power-law decaying function. The
universal behaviour of different observables, especially the thermal and the
order-parameter critical exponents, are computed by Monte-Carlo techniques for
$q=1,2,3$-Potts models for different values of the power-law decaying exponent
$a$. On the basis of our conclusions, which are in agreement with previous
theoretical and numerical results for $q=1$ and $q=2$, we can conjecture the
phase diagram for $q\in [1,4]$. In particular, we establish that the system is
driven to a fixed point at finite or infinite long-range disorder depending on
the values of $q$ and $a$. Finally, we discuss the role of the higher cumulants
of the disorder distribution. This is done by drawning the auxiliary
spin-variables from different statistical models. While the main features of
the phase diagram depend only on the first and second cumulant, we argue, for
the infinite disorder fixed point, that certain universal effects are affected
by the higher cumulants of the disorder distribution.
