  Motivated by recent measurements of the radiative decay rates of the
\emph{P}-wave spin singlet charmonium $h_c$ to the light meson $\eta$ or
$\eta^\prime$ by the BESIII Collaboration, we investigate the decay rates of
these channels at order $\alpha \alpha_s^4$. The photon is radiated mainly from
charm quark pairs in the lowest order Feynman diagrams, since the diagrams
where a photon radiated from light quarks are suppressed by $\alpha_s$ or the
relative charm quark velocity $v$, due to Charge parity conservation. The form
factors of two gluons to $\eta$ or $\eta^\prime$ are employed, which are the
major mechanism for $\eta$ and $\eta^\prime$ productions. $\eta(\eta^\prime)$
is treated as a light cone object when we consider that the parent charmonium
mass is much heavier than that of the final light meson. We obtain the
branching ratio ${\cal B}(h_c\to \gamma\eta^\prime) =
(1.94^{+0.70}_{-0.51})\times 10^{-3}$ in the nonrelativistic QCD approach,
which is in agreement with the BESIII measurement. The prediction of the
branching ratio of $h_c\to \gamma\eta$ is also within the range of experimental
error after including the larger uncertainty of the total decay width
$\Gamma_{h_c}$. The applications of these formulae to the radiative decays to
$\eta(\eta^\prime)$ of the \emph{P}-wave spin singlet bottomonium $h_b(nP)$ are
presented. These studies will shed some light on the $\eta - \eta^\prime$
mixing effects, the flavor SU(3) symmetry breaking, as well as the
nonperturbative dynamics of charmonium and bottomonium.
