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We examine the possibilities of accommodating the muon $g-2$ anomaly reportedby Fermilab in the 2HDM with a discrete $Z_4$ symmetry in which an inert Higgsdoublet field ($H,~A,~H^\pm$) has the lepton flavor violation $\mu$-$\tau$interactions. We study the case of light $H$ (5 GeV $<m_H<$ 115 GeV) and assumethe Yukawa matrices to be real and symmetrical. Considering relevanttheoretical and experimental constraints, especially for the multi-leptonsearches at the LHC, we find the muon $g-2$ anomaly can be explained within$2\sigma$ range in the region of 5 GeV $<m_H<20$ GeV, 130 GeV $<m_A~(m_{H^\pm})<$ 610 GeV, and 0.005 $<\rho<$ 0.014. Meanwhile, the$\chi^2_\tau$ fitting the data of lepton flavour universality in the $\tau$decays approaches to the SM prediction.

muon $g-2$ anomaly and $\mu$-$\tau$-philic Higgs doublet with a light CP-even component