In this talk I will report on a joint project with Matteo Penegini (Genova). The second cohomology of a surface S as mentioned in the title splits up as a sum of two pieces. One piece comes from the Albanese variety. The other piece looks like the cohomology of a K3 surface, which we call a K3 partner X of S. If the surface S is a product-quotient then we can geometrically construct the K3 partner X and an algebraic correspondence that relates the cohomology of S and X. Finally, we prove the Tate and Mumford-Tate conjectures for all surfaces S that lie in the same connected component of the Gieseker moduli space as a product-quotient surface.