  Recently, the Muon g-2 experiment at Fermilab has measured the muon anomalous
magnetic dipole moment (MDM), $a_\mu=(g_\mu-2)/2$, which reported that the new
experimental average increases the tension between experiment and the standard
model (SM) prediction to 4.2$\sigma$. In this work, we reanalyse the muon
anomalous MDM at two-loop level in the $\mu$ from $\nu$ Supersymmetric Standard
Model ($\mu\nu$SSM) combined with the updated experimental average. The
$\mu\nu$SSM can explain the current tension between the experimental
measurement and the SM theoretical prediction for the muon anomalous MDM,
constrained by the 125 GeV Higgs boson mass and decays, the rare decay
$\bar{B}\rightarrow X_s\gamma$ and so on. We also investigate the anomalous MDM
of the electron and tau lepton, $a_e=(g_e-2)/2$ and $a_\tau=(g_\tau-2)/2$, at
two-loop level in the $\mu\nu$SSM. In addition, the 125 GeV Higgs boson decays
to a pair of charged leptons in the $\mu\nu$SSM are also analysed.
