Measurements of the branching fractions of the P-wave charmonium spin-singlet state <i>h_c</i>(<SUP>1</SUP><i>P</i>₁) → <i>h</i><SUP>+</SUP> <i>h</i><SUP>-</SUP> π<SUP>0</SUP>/η

Based on (2712.4 +/- 14.3) x 10(6) psi(3686) events, we investigate four hadronic decay modes of the P-wave charmonium spin-singlet state h(c)(P-1(1)) -> h(+) h(-) pi(0)/eta (h =pi or K) via the process psi(3686) -> pi(0)h(c) at BESIII. The h(c) -> pi(+)pi(0) decay is observed with a significance of 9.6 sigma after taking into account systematic uncertainties. Evidences for h(c) -> K+K-pi(0) and h(c) -> 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(h(c) -> pi(+)pi(-)pi(0)) = (1.36 +/- 0.16 +/- 0.14)x10(-3), B(h(c) -> K+K-pi(0)) = (3.26 +/- 0.84 +/- 0.36) x 10(-4), and B(h(c) -> K+K-eta) = (3.13 +/- 1.08 +/- 0.38) x 10(-4), where the first uncertainties are statistical and the second are systematic. No significant signal of h(c) -> pi(+)pi(-)eta is found, and the upper limit of its decay branching fraction is determined to be B(h(c) -> pi(+)pi(-)eta) < 4.0 x 10(-4) at the 90% confidence level.