Motivated by recent measurements of the radiative decay rates of the $P$-wave spin singlet charmonium ${h}_{c}$ to the light meson $\ensuremath{\eta}$ or ${\ensuremath{\eta}}^{\ensuremath{'}}$ by the BESIII Collaboration, we investigate the decay rates of these channels at order $\ensuremath{\alpha}{\ensuremath{\alpha}}_{s}^{4}$. The photon is radiated mainly from charm quark pairs in the lowest-order Feynman diagrams, because the diagrams where a photon radiated from light quarks are suppressed by ${\ensuremath{\alpha}}_{s}$ or the relative charm quark velocity $v$, due to charge-parity conservation. The form factors of two gluons to $\ensuremath{\eta}$ or ${\ensuremath{\eta}}^{\ensuremath{'}}$, which are the major mechanism for $\ensuremath{\eta}$ and ${\ensuremath{\eta}}^{\ensuremath{'}}$ production, are employed. $\ensuremath{\eta}({\ensuremath{\eta}}^{\ensuremath{'}})$ 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 $\mathcal{B}({h}_{c}\ensuremath{\rightarrow}\ensuremath{\gamma}{\ensuremath{\eta}}^{\ensuremath{'}})=(1.9{4}_{\ensuremath{-}0.51}^{+0.70})\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ in the nonrelativistic QCD approach, which is in agreement with the BESIII measurement. The prediction of the branching ratio of ${h}_{c}\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\eta}$ is also within the range of experimental error after including the larger uncertainty of the total decay width ${\mathrm{\ensuremath{\Gamma}}}_{{h}_{c}}$. The applications of these formulas to the radiative decays to $\ensuremath{\eta}({\ensuremath{\eta}}^{\ensuremath{'}})$ of the $P$-wave spin singlet bottomonium ${h}_{b}(nP)$ are presented. These studies will shed some light on the $\ensuremath{\eta}\ensuremath{-}{\ensuremath{\eta}}^{\ensuremath{'}}$ mixing effects, the flavor SU(3) symmetry breaking, and the nonperturbative dynamics of charmonium and bottomonium.