The processes h(c) -> gamma P (P = eta', eta, pi(0)) are studied with a sample of (27.12 +/- 0.14) x 10(8) psi(3686) events collected by the BESIII detector at the BEPCII collider. The decay h(c) -> 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 +/- 0.55 +/- 0.13 +/- 0.26) x 10(-4), while B(h(c) -> gamma eta ') is measured to be (1.40 +/- 0.11 +/- 0.04 +/- 0.10) x 10(-3), where the first uncertainties are statistical, the second systematic, and the third from the branching fraction of psi(3686) -> pi(0)h(c). The combination of these results allows for a precise determination of R-hc B(h(c)->gamma eta)/B(h(c)->gamma eta '), which is calculated to be (27.0 +/- 4.4 +/- 1.0)%. The results are valuable for gaining a deeper understanding of eta - eta ' mixing, and its manifestation within quantum chromodynamics. No significant signal is found for the decay h(c) -> gamma pi(0), and an upper limit is placed on its branching fraction of B(h(c) -> gamma pi(0)) < 5.0 x 10(-5), at the 90% confidence level.