Using data samples collected at center-of-mass energies of √s=4.009, 4.230, 4.260, and 4.360 GeV with the BESIII detector operating at the BEPCII collider, we perform a search for the process e+e−→γχcJ (J=0, 1, 2)and find evidence for e+e−→γχc1 and e+e−→γχc2 with statistical significances of 3.0σ and 3.4σ, respectively. The Born cross sections σB(e+e−→γχcJ ), as well as their upper limits at the 90% confidence level (C.L.) are determined