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The processes hc→γP(P=η′, η, π0) are studied with a sample of (27.12±0.14)×108 ψ(3686) events collected by the BESIII detector at the BEPCII collider. The decay hc→γη is observed for the first time with the significance of 9.0σ, and the branching fraction is determined to be (3.77±0.55±0.13±0.26)×10−4, while B(hc→γη′) is measured to be (1.40±0.11±0.04±0.10)×10−3, where the first uncertainties are statistical, the second systematic, and the third from the branching fraction of ψ(3686)→π0hc. The combination of these results allows for a precise determination of Rhc=B(hc→γη)B(hc→γη′), which is calculated to be (27.0±4.4±1.0)%. The results are valuable for gaining a deeper understanding of η−η′ mixing, and its manifestation within quantum chromodynamics. No significant signal is found for the decay hc→γπ0, and an upper limit is placed on its branching fraction of B(hc→γπ0)<5.0×10−5, at the 90% confidence level.