  ${\rm U(1)}_{L_\mu - L_\tau} \equiv {\rm U(1)}_X$ model is anomaly free
within the Standard Model (SM) fermion content, and can accommodate the muon
$(g-2)$ data for $M_{Z'} \sim O(10-100)$ MeV and $g_X \sim (4 - 8) \times
10^{-4}$. WIMP type thermal dark matter (DM) can be also introduced for $M_{Z'}
\sim 2 M_{\rm DM}$, if DM pair annihilations into the SM particles occur only
through the $s$-channel $Z'$ exchange. In this work, we show that this tight
correlation between $M_{Z'}$ and $M_{\rm DM}$ can be completely evaded both for
scalar and fermionic DM, if we include the contributions from dark Higgs boson
($H_1$). Dark Higgs boson plays a crucial role in DM phenomenology, not only
for generation of dark photon mass, but also opening new channels for DM pair
annihilations into the final states involving dark Higgs boson, such as dark
Higgs pair as well as $Z' Z'$ through dark Higgs exchange in the $s$-channel,
and co-annihilation into $Z' H_1$ in case of inelastic DM. Thus dark Higgs
boson will dissect the strong correlation $M_{Z'} \sim 2 M_{\rm DM}$, and much
wider mass range is allowed for $U(1)_X$-charged complex scalar and Dirac
fermion DM, still explaining the muon $(g-2)$. We consider both generic
$U(1)_X$ breaking as well as ${\rm U(1)}_X \rightarrow Z_2$ (and also into
$Z_3$ only for scalar DM case).
