  A $Z'$ gauge boson with sub-GeV mass has acquired a significant interest in
phenomenology, particularly in view of the muon $g-2$ anomaly and coherent
elastic neutrino-nucleon scattering. The latter is challenged by the nuclear
recoil energy of a few tens of keV but has been observed by the COHERENT
experiment. To further reconcile the observed excesses in $R(D^{(*)})$ from
semileptonic charmful $B$ decays and in the $W$ boson mass, we investigate a
model with a gauged $U(1)_{L_\mu-L_\tau}$ symmetry and a scalar leptoquark. In
contrast to the mechanism that involves kinetic mixing between the gauge bosons
of $U(1)_{\rm em}$ and $U(1)_{L_\mu-L_\tau}$, we adopt a dynamical symmetry
breaking of $U(1)_{L_\mu-L_\tau}$ by incorporating an additional Higgs doublet.
Through mixing with the $U(1)_{L_\mu-L_\tau}$-charged Higgs doublet, new Higgs
decay channels $h\to Z_1 Z_1/Z_1 Z_2$ occur at percent-level branching ratios,
which could be accessible at the LHC. The $W$-mass anomaly observed by CDF II
can be potentially resolved through the enhancement in the oblique parameter
$T$. Due to the flavored gauge symmetry, the introduced scalar leptoquark
$S^{\frac{1}{3}}=(\bar{3},1,2/3)$ exhibits a unique coupling to the
$\tau$-lepton, offering an explanation for the excesses observed in
$R(D^{(*)})$. Moreover, $\tau \to \mu (Z_1\to ) e^- e^+$ via the resonant light
gauge boson decay can reach the sensitivity of Belle II at an integrated
luminosity of 50 ab$^{-1}$.
