Using 106 ×106 ψ(3686) events collected with the BESIII detector, we measure multipole amplitudes for the decay ψ(3686)→γχc1,2→γγJ/ψ beyond the dominant electric-dipole amplitudes. The normalized magnetic-quadrupole (M2) amplitude for ψ(3686)→γχc1,2→γγJ/ψ and the normalized electric-octupole amplitudes for ψ(3686)→γχc2, χc2→γJ/ψ are determined. The M2 amplitudes for ψ(3686)→γχc1 and χc1,2→γJ/ψ are found to differ significantly from zero and are consistent with theoretical predictions. We also obtain the ratios of M2 contributions of ψ(3686) and J/ψ decays to χc1,2, b21/b22=1.35±0.72 and a21/a22=0.617±0.083, which agree well with theoretical expectations. By considering the multipole contributions of χc1,2, we measure the product branching fractions for the cascade decays ψ(3686)→γχc0,1,2→γγJ/ψ and search for the process ηc(2S)→γJ/ψ through ψ(3686)→γηc(2S). The product branching fraction for ψ(3686)→γχc0→γγJ/ψ is 3σ larger than published measurements, while those of ψ(3686)→γχc1,2→γγJ/ψ are consistent. No significant signal for the decay ψ(3686)→γηc(2S)→γγJ/ψ is observed, and the upper limit of the product branching fraction at the 90% confidence level is determined.