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.
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