Using data samples with an integrated luminosity of 6.4 fb−1 collected by the BESIII detector operating at the BEPCII storage ring, the process of e+e− → γϕJ/ψ is studied. The processes of e+e− → ϕχc1,c2, χc1,c2 → γJ/ψ are observed with a significance of more than 10σ. The s-dependent cross section of e+e− → ϕχc1,c2 is measured between 4.600 and 4.951 GeV, and evidence of a resonance structure is found for the first time in the ϕχc2 process. We also search for the processes of e+e− → γX(4140), γX(4274) and γX(4500) via the γϕJ/ψ final state, but no obvious structures are found. The upper limits on the production cross section times the branching fraction for these processes at the 90% confidence level are reported. [Figure not available: see fulltext.].