  In the model proposed by B\'elanger, Delaunay and Westhoff (BDW), a new
sector consisted of vectorlike fermions and two complex scalars is charged
under an extra Abelian symmetry $U(1)_X$. In this paper, we generalize the BDW
model by introducing the kinetic mixing between the $U(1)_X$ and the standard
model $U(1)_Y$ gauge fields. The new physics contributions to the muon
anomalous magnetic moment and the Wilson coefficients $C_{9,10}^{(')}$ are
obtained analytically. We have explored the free parameter space of the model,
taking into account various constraints on the muon $g-2$ using recent data
from the E989 experiment at Fermilab, the lepton universality violation in
terms of $R_K$ and $R_{K^*}$, and the branching ratios of the semileptonic
decays, $B^+ \rightarrow K^+ \mu^+ \mu^-$ and $B^0 \rightarrow K^{*0} \mu^+
\mu^-$, the LEP and LHC searches for sleptons and $Z'$ boson, as well as the
perturbative requirement. The viable parameter regions of the model are
identified. In the presence of the gauge kinetic mixing term, those regions are
enlarged and significantly deformed in comparison to the case with vanishing
kinetic mixing. In the near future, the E989 experiment with the projected
sensitivity will be able to test significant parts of the currently allowed
parameter regions.
