  Motivated by the growing evidence for lepton flavour universality violation
after the first results from Fermilab's muon $(g-2)$ measurement, we revisit
one of the most widely studied anomaly free extensions of the standard model
namely, gauged $L_{\mu}-L_{\tau}$ model, known to be providing a natural
explanation for muon $(g-2)$. We also incorporate the presence of dark matter
(DM) in this model in order to explain the recently reported electron recoil
excess by the XENON1T collaboration. We show that the same neutral gauge boson
responsible for generating the required muon $(g-2)$ can also mediate
interactions between electron and dark fermions boosted by dark matter
annihilation. The required DM annihilation rate into dark fermion require a
hybrid setup of thermal and non-thermal mechanisms to generate DM relic
density. The tightly constrained parameter space from all requirements remain
sensitive to ongoing and near future experiments, keeping the scenario very
predictive.
