L µ − L τ 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 − → γ + / E process through the kinetic mixing with the photon, where the missing energy / E is from the Z ′ →νν decays. We study the phenomenological implications when a pair of singlet vector-like leptons carrying different L µ − L τ charges are included, and a complex singlet scalar (φ S ) is introduced to accomplish the spontaneous U (1) Lµ−Lτ symmetry breaking. It is found that the extension leads to several phenomena of interest, including (i) branching ratio (BR) for h → µτ can be of the order of 10 −3 ; (ii) φ S -mediated muon g − 2 can be of the order of 10 × 10 −10 ; (iii) BR for τ → µφ * S → µZ ′ 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 τ → (3µ + / E, 5µ) 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.