  We illustrate and discuss the phenomenology of a model featuring a two-Higgs
doublet sector augmented by two $SU(2)$ singlet scalars. The gauge symmetry
group is extended as well with a $U(1)_{B_{i}-L_{i}}$ component whose
spontaneous breaking leads to the gauge boson which has an important effect in
the muon (g-2). A global PQ symmetry is introduced upon its breaking we have
the axion particle which is also DM in our work. In particular, we have
focussed on Type-X and Type-II 2HDM models and found out that (g-2) can not be
explained only by the scalar sector for Type-II 2HDM mainly due to stringent
constraint from $b \rightarrow s \gamma$ resulted in $M_{H^{\pm}} > 800$ GeV.
For Type-II 2HDM, we can have axion coupling with the gluons which generates
the axion potential and possible explanation for the strong CP problem. The
proposed model accommodates neutrino masses via the Type-I see-saw mechanism
with an upper bound on the right-handed neutrino mass 1 GeV (1 TeV) for Type-II
(Type-X) 2HDM due to the presence of Planck scale suppressed operators.
Moreover, we also have additional scalars which affect the oblique parameters
and hence the W-boson mass which leads us to explain the W-boson mass observed
at CDF-II detector. The most stringent constraints on the masses and quartic
couplings come from the perturbativity and potential bound from below
conditions which leads to fine-tuning among the parameters in part of the
parameter space. Finally, we discuss the possible detection prospects of the
axion DM and the additional gauge boson.
