In this talk, we will introduce a particular kind of holomorphic correspondences, called covering correspondences. These are induced by a rational map \(R\) by the relation \(R(z)=R(w)\), deleting the obvious association of \(z\) with itself. The composition of two covering correspondences generalizes the family studied by Bullett et al, which falls into the non modular - weakly modular gap, and for which we know the equidistribution towards the past and the future, as well as that of periodic points. We will talk about some properties of composition of two covering correspondences; in particular, we will show it also falls into the non modular - weakly modular gap.