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The gauged $U(1)_{L_\mu-L_\tau}$ extension of the Standard Model is a verysimple framework that can alleviate the tension in muon anomalous magneticdipole moment, reinforced by the recent Fermilab measurement. We exploreexperimental probes of the $(g-2)_\mu$ target with a general treatment ofkinetic mixing between the $Z'$ gauge boson and the photon. The physical valueof the kinetic mixing depends on a free parameter of the model and energy scaleof a process. We find neutrino constraints on the $(g-2)_\mu$ target includingBorexino, CE$\nu$NS, and white dwarfs are sensitive to this freedom and can belifted if the kinetic mixing lies in proximity of zero at low momentumtransfer. As a further step, we explore $L_\mu-L_\tau$ charged dark matter witha thermal origin and show that the same scenario of kinetic mixing can relaxexisting direct detection constraints and predict novel recoil energydependence in the upcoming searches. Future joint effort of neutrino and darkmatter experiments and precision spectral measurement will be the key to testsuch a theory.

General Kinetic Mixing in Gauged $U(1)_{L_\mu-L_\tau}$ Model for Muon $g-2$ and Dark Matter