  Observation of $\tau^-\to\eta^{(\prime)}\pi^-\nu_\tau $ decays at Belle-II
would indicate either a manifestation of isospin symmetry breaking or genuine
second class currents (SCC) effects. The corresponding radiative
$\tau^-\to\eta^{(\prime)}\pi^-\nu_\tau \gamma$ decay channels are not
suppressed by $G$-parity considerations and may represent a serious background
in searches of SCC in the former. We compute the observables associated to
these radiative decays using Resonance Chiral Lagrangians and conclude that
vetoing photons with $E_\gamma>100$ MeV should get rid of this background in
the Belle-II environment searching for the $\tau^-\to\eta \pi^-\nu_\tau $
channel. Similar considerations hold inconclusive for decays involving the
$\eta^\prime$ given the theory uncertainties in the prediction of the
$\tau^-\to\eta^\prime\pi^-\nu_\tau$ branching ratio. Still, additional
kinematics-based cuts should be able to suppress this background in the
$\eta^\prime$ case to a negligible level.
