  We study the implications of the recent anomalies in the $W$-boson mass and
the anomalous magnetic moment of the muon on $U(1)_{L_\mu-L_\tau}$ model. We
show that the introduction of vector-like leptons which mix with muon can solve
both anomalies. Contrary to the conventional wisdom the electroweak scale
$Z'$-boson is allowed without conflict with the trident neutrino production
experiments.
