  Recently, the BESIII Collaboration reported two new decay processes
$h_c(1P)\to \gamma \eta$ and $\gamma \eta^\prime$. Inspired by this
measurement, we propose to study the radiative decays of $h_c$ via intermediate
charmed meson loops in an effective Lagrangian approach. With the acceptable
cutoff parameter range, the calculated branching ratios of $h_c(1P)\to \gamma
\eta$ and $\gamma \eta^\prime$ are orders of $10^{-4}\sim 10^{-3}$ and $10^{-3}
\sim 10^{-2}$, respectively. The ratio $R_{h_c}= \mathcal{B}( h_c\to \gamma
\eta )/\mathcal{B}( h_c\to \gamma \eta^\prime )$ can reproduce the experimental
measurements with the commonly acceptable $\alpha$ range. This ratio provide us
some information on the $\eta-\eta^\prime$ mixing, which may be helpful for us
to test SU(3)-flavor symmetries in QCD.
