  Models with an Axion Like Particle (ALP) can provide an explanation for the
discrepancy between experimental measurement of the muon anomalous magnetic
moment $(g-2)_\mu$ and the Standard Model prediction. This explanation relies
on the couplings of the ALP to the muon and the photon. We also include more
general couplings to the electroweak gauge bosons and incorporate them in the
calculations up to the 2-loop order. We investigate the existing experimental
constraints and find that they do not rule out the ALP model under
consideration as a possible explanation for the $(g-2)_\mu$ anomaly. At the
same time, we find the future Tera-$Z$ and Higgs factories, such as the CEPC
and FCC-ee, can completely cover the relevant parameter space through searches
with final states $(\gamma \gamma) \gamma$, $(\mu^+ \mu^-) \gamma$ and $(\mu^+
\mu^-) \mu^+ \mu^-$.
