  We generalize the recently introduced TRILEX approach (TRiply Irreducible
Local EXpansion) to superconducting phases. The method treats simultaneously
Mott and spin-fluctuation physics using an Eliashberg theory supplemented by
local vertex corrections determined by a self-consistent quantum impurity
model. We show that, in the two-dimensional Hubbard model, at strong coupling,
TRILEX yields a $d$-wave superconducting dome as a function of doping. Contrary
to the standard cluster dynamical mean field theory (DMFT) approaches, TRILEX
can capture $d$-wave pairing using only a single-site effective impurity model.
We also systematically explore the dependence of the superconducting
temperature on the bare dispersion at weak coupling, which shows a clear link
between strong antiferromagnetic (AF) correlations and the onset of
superconductivity. We identify a combination of hopping amplitudes particularly
favorable to superconductivity at intermediate doping. Finally, we study within
$GW$+EDMFT the low-temperature $d$-wave superconducting phase at strong
coupling in a region of parameter space with reduced AF fluctuations.
