  The astrophysical origin of $r$-process nuclei remains a long-standing
mystery. Although some astrophysical scenarios show some promise, many
uncertainties involved in both the astrophysical conditions and in the nuclear
properties far from the $\beta$-stability have inhibited us from understanding
the nature of the $r$-process. The purpose of the present paper is to examine
the effects of the newly-derived microscopic Hartree-Fock-Bogoliubov (HFB) mass
formulas on the $r$-process nucleosynthesis and analyse to what extent a
solar-like $r$-abundance distribution can be obtained. The $r$-process
calculations with the HFB-2 mass formula are performed, adopting the
parametrized model of the prompt explosion from a collapsing O-Ne-Mg core for
the physical conditions and compared with the results obtained with the HFB-7
and droplet-type mass formulas. Due to its weak shell effect at the neutron
magic numbers in the neutron-rich region, the microscopic mass formulas (HFB-2
and HFB-7) give rise to a spread of the abundance distribution in the vicinity
of the $r$-process peaks ($A = 130$ and 195). While this effect resolves the
large underproduction at $A \approx 115$ and 140 obtained with droplet-type
mass formulas, large deviations compared to the solar pattern are found near
the third $r$-process peak. It is shown that a solar-like $r$-process pattern
can be obtained if the dynamical timescales of the outgoing mass trajectories
are increased by a factor of about 2-3, or if the $\beta$-decay rates are
systematically increased by the same factor.
