Very recently, the Fermi-Lab reported the new experimental combined results on the magnetic momentum of muon with a 4.2$sigma$ discrepancy compared with the expectation of the Standard Model cite{Fermi_Lab}. A new light gauge boson $X$ in the $L_{mu}-L_{tau}$ model provides a good explanation for the $g-2$ anomaly. A Dirac fermion dark matter with a large $L_{mu}-L_{tau}$ charge can explain both the $g-2$ anomaly and the dark matter relic density cite{Asai_2021}. In this work, we focus on the case that the mass of the dark matter is larger than the mass of muon (i.e. $m_{Psi} > m_{mu}$) for which the channel $Psi Psi rightarrow mu^- mu^+$ opens. Although the cross section $(sigma v)_{mu^{-}mu^{+}}$ is smaller by a factor of $1/q_{Psi}^2$ ($q_{Psi}$ represents the $L_{mu}-L_{tau}$ charge of the dark matter) compared with the channel $PsiPsi rightarrow XX rightarrow u ubar{ u}bar{ u}$, the resulting secondary electrons and positrons could imprint on their spectra above GeV energies due to the reacceleration effect of cosmic ray propagation. We use the AMS-02 measurements of electrons and positrons to constrain the annihilation cross section of the channel $PsiPsi rightarrow mu^{-}mu^{+}$, which rules out part of the parameter space of the large $L_{mu}-L_{tau}$ charged dark matter model to account for the muon $g-2$ anomaly.
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