  Doping a Mott insulator gives rise to unconventional superconducting
correlations. Here we address the interplay between d-wave superconductivity
and Mott physics using the two-dimensional Hubbard model with cellular
dynamical mean-field theory on a $2\times2$ plaquette. Our approach is to study
superconducting correlations from the perspective of a cluster quantum impurity
model embedded in a self-consistent bath. At the level of the cluster, we
calculate the probabilities of the possible cluster electrons configurations.
Upon condensation we find an increased probability that cluster electrons
occupy a four-electron singlet configuration, enabling us to identify this type
of short-range spin correlations as key to superconducting pairing. The
increased probability of this four-electron singlet comes at the expense of a
reduced probability of a four-electron triplet with no significant probability
redistribution of fluctuations of charges. This allows us to establish that
superconductivity at the level of the cluster primarily involves a
reorganisation of short-range spin correlations rather than charge
correlations. We gain information about the bath by studying the spectral
weight of the hybridization function. Upon condensation, we find a transfer of
spectral weight leading to the opening of a superconducting gap. We use these
insights to interpret the signatures of superconducting correlations in the
density of states of the system and in the zero-frequency spin susceptibility.
