The $e^+e^-togammachi_{cJ}$ ($J=0,1,2$) processes are studied at center-of-mass energies ranging from 3.773 to 4.600~GeV, using a total integrated luminosity of 19.3~fb$^{-1}$ $e^+e^-$ annihilation data accumulated with the BESIII detector at BEPCII. We observe for the first time $e^+e^-togamma chi_{c1,c2}$ signals at $sqrt{s}=$ 4.180~GeV with statistical significances of 7.6$sigma$ and 6.0$sigma$, respectively. The production cross section of $e^+e^-togamma chi_{c1,c2}$ at each center-of-mass energy is also measured. We find that the line shape of the $e^+e^-togamma chi_{c1}$ cross section can be described with conventional charmonium states $psi(3686)$, $psi(3770)$, $psi(4040)$, $psi(4160)$. Compared with this, for the $e^+e^-togamma chi_{c2}$ channel, one more additional resonance is added to describe the cross section line shape. Its mass and width are measured to be $M=4371.7pm7.5pm1.8$~MeV/$c^2$ and $Gamma^{tot}=51.1pm17.6pm1.9$~MeV, where the first uncertainties are statistical and the second systematic. The significance of this resonance is estimated to be 5.8$sigma$, and its parameters agree with the $Y(4360)$ resonance previously reported in $e^+e^-topi^+pi^-psi(3686)$, and the $Y(4390)$ in $e^+e^-topi^+pi^-h_c$ within uncertainties. No significant signal for the $e^+e^-togamma chi_{c0}$ process is observed, and the upper limits of Born cross sections $sigma_{B}(e^+e^-togamma chi_{c0})$ at 90% confidence level are reported.
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