  The discrepancy between the muon $g-2$ measurement and the Standard Model
prediction points to new physics around or below the weak scale. It is
tantalizing to consider the loop effects of a heavy axion (in the general
sense, also known as an axion-like particle) coupling to leptons and photons as
an explanation for this discrepancy. We provide an updated analysis of the
necessary couplings, including two-loop contributions, and find that the new
physics operators point to an axion decay constant on the order of 10s of GeV.
This poses major problems for such an explanation, as the axion couplings to
leptons and photons must be generated at low scales. We outline some
possibilities for how such couplings can arise, and find that these scenarios
predict new charged matter at or below the weak scale and new scalars can mix
with the Higgs boson, raising numerous phenomenological challenges. These
scenarios also all predict additional contributions to the muon $g-2$ itself,
calling the initial application of the axion effective theory into question. We
conclude that there is little reason to favor an axion explanation of the muon
$g-2$ measurement relative to other models postulating new weak-scale matter.
