  We investigate the minimal $U(1)_{L_\mu-L_\tau}$ model with extra heavy
vector-like leptons or charged scalars. By studying the kinetic mixing between
$U(1)_{L_\mu-L_\tau}$ gauge boson $Z^\prime$ and standard model photon, which
is absent at tree level and will arise at one loop level due to $\mu$, $\tau$
and new heavy charged leptons or scalars, the interesting behavior is shown. It
can provide possibility for visible signatures of new heavy particles. We
propose to search for $Z^\prime$ at electron collider experiments, such as
Belle II, BESIII and future Super Tau Charm Factory (STCF), using the
monophoton final state. The parameter space of $Z^\prime$ is probed, and
scanned by its gauge coupling constant $g_{Z^\prime}$ and mass $m_{Z^\prime}$.
We find that electron colliders have sensitivity to the previously unexplored
parameter space for $Z^\prime$ with MeV-GeV mass. Future STCF experiments with
$\sqrt s=2-7$ GeV can exclude the anomalous muon magnetic moment favored area
when $m_{Z^\prime}<5$ GeV with the luminosity of 30 ab$^{-1}$. For
$m_{Z^\prime} < 2m_\mu$, $g_{Z^\prime}$ can be down to $4.2\times 10^{-5}$ at 2
GeV STCF.
