  The processes $h_c\to\gamma P(P = \eta^\prime,~\eta,~\pi^0)$ are studied with
a sample of $(27.12\pm0.14)\times10^{8}$ $\psi(3686)$ events collected by the
BESIII detector at the BEPCII collider. The decay $h_{c}\rightarrow\gamma\eta$
is observed for the first time with the significance of $9.0\,\sigma$, and the
branching fraction is determined to be
$(3.77\pm0.55\pm0.13\pm0.26)\times10^{-4}$, while
$\mathscr{B}(h_{c}\rightarrow\gamma\eta^\prime)$ is measured to be
$(1.40\pm0.11\pm0.04\pm0.10)\times10^{-3}$, where the first uncertainties are
statistical, the second systematic, and the third from the branching fraction
of $\psi(3686)\rightarrow\pi^{0}h_c$. The combination of these results allows
for a precise determination of
$R_{h_c}=\frac{\mathscr{B}(h_c\rightarrow\gamma\eta)}{\mathscr{B}(h_c\rightarrow\gamma\eta^\prime)}$,
which is calculated to be $(27.0\pm4.4\pm1.0)\%$. The results are valuable for
gaining a deeper understanding of $\eta-\eta^\prime$ mixing, and its
manifestation within quantum chromodynamics. No significant signal is found for
the decay $h_c\rightarrow\gamma\pi^{0}$, and an upper limit is placed on its
branching fraction of
$\mathscr{B}(h_c\rightarrow\gamma\pi^{0})<5.0\times10^{-5}$, at the 90$\%$
confidence level.
