  Minimal gauged $U(1)_{L_\mu-L_\tau}$ model has long been known to be able to
explain the tension between the theoretical and experimental values of the muon
magnetic moment. It has been explored and tested extensively, pushing the
viable parameter space into a very tight corner. Further, embedding the
$U(1)_{L_\mu-L_\tau}$ model in a supersymmetric (SUSY) framework has been shown
to relax some of these constraints and has recently been shown to explain the
electron anomalous magnetic moment as well. In this model, the logarithm of the
mass ratio of third to second generation (s)leptons control the non-negligible
kinetic mixing and may crucially alter many of the constraints. We confront
both the non-SUSY and SUSY versions of this class of models with the CsI(2017),
the recently released CENNS10 data from the liquid Argon detector as well as
the updated CsI(2020) data of the COHERENT experiment. We use the recoil energy
and timing binned data from CsI(2017) and the energy, time and PSD binned data
from CENNS10 to find estimates for the model parameters in a likelihood
maximization test. We also show updated exclusions using all of the above data
from the COHERENT Collaboration, as well as projected exclusions from the
ongoing Coherent CAPTAIN-Mills (CCM) experiment. The $(g-2)_\mu$ favored values
of the $U(1)_{L_\mu-L_\tau}$ gauge coupling that are still unconstrained
overlaps with the estimates from COHERENT data within $1\sigma$. The combined
COHERENT data is found to prefer the presence of the $U(1)_{L_\mu-L_\tau}$
gauge boson over the SM at $\sim1.4\sigma$. The global minima of a chi-square
deviation function using CsI(2020) as well as CENNS10 total counts has
significant overlap with the $(g-2)_{\mu,e}$ favored parameter space in the
context of SUSY $L_{\mu}-L_{\tau}$ model.
