We report on an analysis of the decay J/ψ→γπ0η′ using a sample of (1310.6±7.0)×106J/ψ events collected with the BESIII detector. We search for the CP-violating process ηc→π0η′ and a dark gauge boson U′ in J/ψ→U′η′,U′→γπ0,π0→γγ. No evidence of an ηc signal is observed in the π0η′ invariant-mass spectrum and the upper limit of the branching fraction is determined to be 5.6×10-5 at the 90% confidence level. We also find no evidence of U′ production and set upper limits at the 90% confidence level on the product branching fraction B(J/ψ→U′η′)×B(U′→π0γ) in the range between (0.8-6.5)×10-7 for 0.2≤mU′≤2.1 GeV/c2. In addition, we study the process J/ψ→ωη′ with ω→γπ0. The branching fraction of J/ψ→ωη′ is found to be (1.87±0.09±0.12)×10-4, where the first uncertainty is statistical and the second is systematic, with a precision that is improved by a factor of 1.4 over the previously published BESIII measurement.