This dissertation shows the first evidence of the process e+e− → γηc(1S) using data collected by the BESIII experiment operating at BEPCII. This process can be used as a probe to study the nature of recently discovered charmonium-like Y states between 4.0 and 4.6 GeV, including the Y (4260) and Y (4360). Data collected at six center-of-mass energies are analyzed, namely: 4.01, 4.23, 4.26, 4.36, 4.42, and 4.60 GeV, corresponding to a total integrated luminosity of 4.6 fb−1. We measure the Born cross section, σE(e+e− → γηc(1S)), at each energy using a combination of twelve ηc(1S) decay channels. Because the significance of the signal is marginal at each energy (≤ 3.0σ), we also combine all six energies under various assumptions for the energy-dependence of the cross section. If a Y (4260) is assumed, we measure σ4.26(e+e− → γηc(1S)) = 2.11 ± 0.49(stat.) ± 0.33(syst.) pb with a significance of 4.2σ. With our current statistics we are unable to distinguish the Y (4260) process from others.