Measurements of the branching fractions of the $P$-wave charmonium spin-singlet state $h_c(^1P_1) \to h^+ h^-π^0/η$
Based on $(2712.4\pm 14.3)\times10^{6}$ $ψ(3686)$ events, we investigate four hadronic decay modes of the $P$-wave charmonium spin-singlet state $h_c(^1P_1) \to h^+ h^- π^0/η$ ($h=π$ or $K$) via the process $ψ(3686) \to π^{0}h_c$ at BESIII. The $h_c \to π^+ π^- π^0$ decay is observed with a significance of 9.6$σ$ after taking into account systematic uncertainties. Evidences for $h_c \to K^+ K^- π^0$ and $h_c \to K^+ K^- η$ are found with significances of $3.5σ$ and $3.3σ$, respectively, after considering the systematic uncertainties. The branching fractions of these decays are measured to be $\mathcal{B}(h_c \to π^+ π^- π^0)=(1.36\pm0.16\pm0.14)\times10^{-3}$, $\mathcal{B}(h_c \to K^+ K^- π^0)=(3.26\pm0.84\pm0.36)\times10^{-4}$, and $\mathcal{B}(h_c \to K^+ K^- η)=(3.13\pm1.08\pm0.38)\times10^{-4}$, where the first uncertainties are statistical and the second are systematic. No significant signal of $h_c\toπ^+π^-η$ is found, and the upper limit of its decay branching fraction is determined to be $\mathcal{B}(h_c\toπ^+π^-η) < 4.0 \times 10^{-4}$ at 90% confidence level.