Measurements of the branching fractions of the $P$-wave charmonium spin-singlet state ${h}_{c}\left({}^{1}{P}_{1}\right)\to {h}^{+}{h}^{-}{\pi }^{0}/\eta $

Based on $\left(2712.4±14.3\right)×{10}^{6}\text{}\text{}\psi \left(3686\right)$ events, we investigate four hadronic decay modes of the $P$-wave charmonium spin-singlet state ${h}_{c}{\left(}^{1}{P}_{1}\right)\to {h}^{+}{h}^{-}{\pi }^{0}/\eta $ ($h=\pi $ or $K$) via the process $\psi \left(3686\right)\to {\pi }^{0}{h}_{c}$ at BESIII. The ${h}_{c}\to {\pi }^{+}{\pi }^{-}{\pi }^{0}$ decay is observed with a significance of $9.6\sigma $ after taking into account systematic uncertainties. Evidences for ${h}_{c}\to {K}^{+}{K}^{-}{\pi }^{0}$ and ${h}_{c}\to {K}^{+}{K}^{-}\eta $ are found with significances of $3.5\sigma $ and $3.3\sigma $, respectively, after considering the systematic uncertainties. The branching fractions of these decays are measured to be $B\left({h}_{c}\to {\pi }^{+}{\pi }^{-}{\pi }^{0}\right)=\left(1.36±0.16±0.14\right)×{10}^{-3}$, $B\left({h}_{c}\to {K}^{+}{K}^{-}{\pi }^{0}\right)=\left(3.26±0.84±0.36\right)×{10}^{-4}$, and $B\left({h}_{c}\to {K}^{+}{K}^{-}\eta \right)=\left(3.13±1.08±0.38\right)×{10}^{-4}$, where the first uncertainties are statistical and the second are systematic. No significant signal of ${h}_{c}\to {\pi }^{+}{\pi }^{-}\eta $ is found, and the upper limit of its decay branching fraction is determined to be $B\left({h}_{c}\to {\pi }^{+}{\pi }^{-}\eta \right)<4.0×{10}^{-4}$ at the 90% confidence level.