  We propose an extra $U(1)_X$ model with an alternative charge assignment for
right-handed right-handed neutrinos. The type-II seesaw mechanism by a triplet
Higgs field is promising for neutrino mass generation because of the
alternative charge assignment. The small vacuum expectation value (VEV) of an
additional Higgs doublet naturally leads to a very small VEV of the triplet
Higgs field, and as a result, the smallness of neutrino mass can be understood.
With the minimal Higgs field for $U(1)_X$ with the charge $1$, right-handed
neutrinos are candidates for Dirac dark matter (DM) and dark radiation (DR). We
have derived and imposed the LHC bound, the DR constraint and the bound from DM
direct searches in the wide range of parameter space. Among various $U(1)_X$
choices, the DM direct search bound is found to be weakest for $U(1)_R$ where
the constraints from thermal DM and non-negligible DR can be compatible. Such a
number of the effective neutrino species would be interesting from the
viewpoint of the so-called Hubble tension.
