We study the evolution of the superconducting phases in a model with competing short-range attractive (W) and on-site repulsive (U) interactions. The influence of the correlations on the phase diagram of $s\ensuremath{-}$ and d-wave superconductors is studied by means of various approximations. The knowledge of the normal state correlation effects allows the tracking of the influence of U on the superconductivity in correlated systems. We have found the qualitatively different response of the superconducting phases in systems with
W<Ucr, when the system ceases to superconduct for large enough U, and in the limit of strong
W>Ucr, when the system remains superconducting for arbitrarily large U.
We present a controlled perturbative approach to the low-temperature phase diagram of highly inhomogeneous Hubbard models in the limit of small coupling, ${t}^{\ensuremath{'}}$, between clusters. We apply this to the dimerized and checkerboard models. The dimerized model is found to behave like a doped semiconductor, with a Fermi-liquid groundstate with parameters (e.g., the effective mass) which are smooth functions of the Hubbard interaction,
U. By contrast, the checkerboard model has a Fermi liquid phase at large
U>Uc=4.58, a
d-wave superconducting state with a full gap for
Uc>U>0, and a narrow strip of an intermediate
d-wave superconducting phase with gapless ``nodal'' quasiparticles for $\ensuremath{\mid}U\ensuremath{-}{U}_{c}\ensuremath{\mid}<\mathcal{O}({t}^{\ensuremath{'}})$.