  We study a minimal extension of the Standard Model by introducing three
right-handed neutrinos and a new scotogenic scalar doublet, in which the mass
splittings between neutral and charged components are responsible for the
$W$-boson mass newly measured by the CDF collaboration. This model can not only
generate non-vanishing Majorana neutrino masses via the interaction of
right-handed neutrinos and scotogenic scalars, but also explain the Universe's
missing matter in the form of FIMP dark matter. We also study the influence of
the mass splitting on the first order electroweak phase transition, and find
that it can further enhance the transition strength and thus induce
gravitational waves during the phase transition, which may be detected in the
forthcoming detectors such as U-DECIGO.
