  We amend a recent dispersive analysis of the anomalous $\eta$ decay process
$\eta\to\pi^+\pi^-\gamma$ by the effects of the $a_2$ tensor meson, the
lowest-lying resonance that can contribute in the $\pi\eta$ system. While the
net effects on the measured decay spectrum are small, they may be more
pronounced for the analogous $\eta'$ decay. There are nonnegligible
consequences for the $\eta$ transition form factor, which is an important
quantity for the hadronic light-by-light scattering contribution to the muon's
anomalous magnetic moment. We predict total and differential cross sections, as
well as a marked forward-backward asymmetry, for the crossed process
$\gamma\pi^-\to\pi^-\eta$ that could be measured in Primakoff reactions in the
future.
