  The nucleon-nucleon correlation between nucleons leads to the Fermi surface
depletion measured by a $Z$-factor in momentum distribution of dense nuclear
matter. The roles of the Fermi surface depletion effect ($Z$-factor effect) and
its quenched neutron triplet superfluidity of nuclear matter in viscosity and
hence in the gravitational-wave-driven $r$-mode instability of neutron stars
(NSs) are investigated. The bulk viscosity is reduced by both the two effects,
especially the superfluid effect at low temperatures which is also able to
reduce the inferred core temperature of NSs. Intriguingly, due to the neutron
superfluidity, the core temperature of the NSs in known low-mass X-ray binaries
(LMXBs) are found to be clearly divided into two groups: high and low
temperatures which correspond to NSs with short and long recurrence times for
nuclear-powered bursts respectively. Yet, a large number of NSs in these LMXBs
are still located in the $r$-mode instability region. If the density-dependent
symmetry energy is stiff enough, the occurence of direct Urca process reduces
the inferred core temperature by about one order of magnitude. Accordingly, the
contradiction between the predictions and observations is alleviated to some
extent, but some NSs are still located inside the unstable region.
