The $P$-wave charmonium decays ${h}_{c}\ensuremath{\rightarrow}\ensuremath{\gamma}{\ensuremath{\eta}}^{(\ensuremath{'})}$ are revisited by taking into account relativistic corrections. The decay amplitudes are derived in the Bethe-Salpeter formalism, in which the involved one-loop integrals are evaluated analytically. Intriguingly, from both the quark-antiquark content and the gluonic content of ${\ensuremath{\eta}}^{(\ensuremath{'})}$, the relativistic corrections make significant contributions to the decay rates of ${h}_{c}\ensuremath{\rightarrow}\ensuremath{\gamma}{\ensuremath{\eta}}^{(\ensuremath{'})}$. By comparison with the leading-order contributions from the quark-antiquark content (one-loop level), the ones from the gluonic content (tree level) are also important, which is compatible with the conclusion obtained without relativistic corrections. Usually, for the $\ensuremath{\eta}$ production processes, the predicted branching ratios are sensitive to the angle of $\ensuremath{\eta}\ensuremath{-}{\ensuremath{\eta}}^{\ensuremath{'}}$ mixing. As an illustration, using the Feldmann-Kroll-Stech result about the mixing angle $\ensuremath{\phi}=39.3\ifmmode^\circ\else\textdegree\fi{}\ifmmode\pm\else\textpm\fi{}1.0\ifmmode^\circ\else\textdegree\fi{}$ as input, we find that the predicted ratio ${R}_{{h}_{c}}=\mathcal{B}({h}_{c}\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\eta})/\mathcal{B}({h}_{c}\ensuremath{\rightarrow}\ensuremath{\gamma}{\ensuremath{\eta}}^{\ensuremath{'}})$ is much smaller than the experiment measurement, while, with $\ensuremath{\phi}=33.5\ifmmode^\circ\else\textdegree\fi{}\ifmmode\pm\else\textpm\fi{}0.9\ifmmode^\circ\else\textdegree\fi{}$ extracted from the asymptotic limit of the ${\ensuremath{\gamma}}^{*}\ensuremath{\gamma}\ensuremath{-}{\ensuremath{\eta}}^{\ensuremath{'}}$ transition form factor, we obtain ${R}_{{h}_{c}}=30.3%$, consistent with ${R}_{{h}_{c}}^{\mathrm{exp}}=(30.7\ifmmode\pm\else\textpm\fi{}11.3\ifmmode\pm\else\textpm\fi{}8.7)%$. As a cross-check, the mixing angle $\ensuremath{\phi}=33.8\ifmmode^\circ\else\textdegree\fi{}\ifmmode\pm\else\textpm\fi{}2.5\ifmmode^\circ\else\textdegree\fi{}$ is extracted by employing the ratio ${R}_{{h}_{c}}$, and a brief discussion on the difference in the determinations of $\ensuremath{\phi}$ is given.