  The gauged $U(1)_{L_\mu-L_\tau}$ extension of the Standard Model is a very
simple framework that can alleviate the tension in muon anomalous magnetic
dipole moment, reinforced by the recent Fermilab measurement. We explore
experimental probes of the $(g-2)_\mu$ target with a general treatment of
kinetic mixing between the $Z'$ gauge boson and the photon. The physical value
of the kinetic mixing depends on a free parameter of the model and energy scale
of a process. We find neutrino constraints on the $(g-2)_\mu$ target including
Borexino, CE$\nu$NS, and white dwarfs are sensitive to this freedom and can be
lifted if the kinetic mixing lies in proximity of zero at low momentum
transfer. As a further step, we explore $L_\mu-L_\tau$ charged dark matter with
a thermal origin and show that the same scenario of kinetic mixing can relax
existing direct detection constraints and predict novel recoil energy
dependence in the upcoming searches. Future joint effort of neutrino and dark
matter experiments and precision spectral measurement will be the key to test
such a theory.
