  The interplay of non-Fermi liquid and superconductivity born out of strong
dynamical interactions is at the heart of the physics of unconventional
superconductivity. As a solvable platform of the strongly correlated
superconductors, we study the pairing instabilities of the
Yukawa-Sachdev-Ye-Kitaev (Yukawa-SYK) model, which describes spin-1/2 fermions
coupled to bosons by the random, all-to-all, spin-independent and dependent
Yukawa interactions. In contrast to the previously studied models, the random
Yukawa couplings are sampled from a collection of Gaussian ensembles whose
variances follow a continuous distribution rather than being fixed to a
constant. By tuning the analytic behaviour of the distribution, we could
control the fermion incoherence to systematically examine various normal states
ranging from the Fermi liquid to non-Fermi liquids that are different from the
conformal solution of the SYK model with a constant variance. Using the
linearised Eliashberg theory, we show that the onset of the unconventional
spin-triplet pairing is preferred with the spin-dependent interactions while
all pairing channels show instabilities with the spin-independent interactions.
Although the interactions shorten the lifetime of the fermions in the non-Fermi
liquid, the same interactions also dress the bosons to strengthen the tendency
to pair the incoherent fermions. As a consequence, the onset temperature $T_c$
of the pairing is enhanced in the non-Fermi liquid compared to the case of the
Fermi liquid.
