  $L_\mu -L_\tau$ gauge boson ($Z'$) with a mass in the MeV to GeV region can
resolve not only the muon $g-2$ excess, but also the gap in the high-energy
cosmic neutrino spectrum at IceCube. It was recently proposed that such a light
gauge boson can be detected during the Belle II experiment with a luminosity of
50 ab$^{-1}$ by the $e^+ e^- \to \gamma +\slashed{E}$ process through the
kinetic mixing with the photon, where the missing energy $\slashed{E}$ is from
the $Z' \to \bar\nu \nu$ decays. We study the phenomenological implications
when a pair of singlet vector-like leptons carrying different $L_\mu - L_\tau$
charges are included, and a complex singlet scalar ($\phi_S$) is introduced to
accomplish the spontaneous $U(1)_{L_\mu -L_\tau}$ symmetry breaking. It is
found that the extension leads to several phenomena of interest, including (i)
branching ratio (BR) for $h\to \mu \tau$ can be of the order of $10^{-3}$; (ii)
$\phi_S$-mediated muon $g-2$ can be of the order of $10\times 10^{-10}$; (iii)
BR for $\tau \to \mu \phi^*_S\to \mu Z'Z'$ can be $10^{-8}$, and (iv) kinetic
mixing between the $Z'$ boson and photon is sensitive to the relative heavy
lepton masses. The predicted BRs for $\tau\to (3\mu+\slashed{E}, 5\mu$) through
the leptonic $Z'$ decays can reach a level of $10^{-9}$, in which the results
fall within the sensitivity of the Belle II in the search for the rare tau
decays.
