Evidence for $e^+e^-togammachi_{c1, 2}$ at center-of-mass energies from 4.009 to 4.360 GeV

Using data samples collected at center-of-mass energies of $sqrt{s}$ = 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 $e^+e^-togammachi_{cJ}$ $(J = 0, 1, 2)$ and find evidence for $e^+e^-togammachi_{c1}$ and $e^+e^-togammachi_{c2}$ with statistical significances of 3.0$sigma$ and 3.4$sigma$, respectively. The Born cross sections $sigma^{B}(e^+e^-togammachi_{cJ})$, as well as their upper limits at the 90% confidence level are determined at each center-of-mass energy.