  The $e^+e^-\to\gamma\chi_{cJ}$ ($J=0,1,2$) 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
$e^+e^-\to\gamma \chi_{c1,c2}$ signals at $\sqrt{s}=$ 4.180 GeV with
statistical significances of 7.6$\sigma$ and 6.0$\sigma$, respectively. The
production cross section of $e^+e^-\to\gamma \chi_{c1,c2}$ at each
center-of-mass energy is also measured. We find that the line shape of the
$e^+e^-\to\gamma \chi_{c1}$ cross section can be described with conventional
charmonium states $\psi(3686)$, $\psi(3770)$, $\psi(4040)$, $\psi(4160)$.
Compared with this, for the $e^+e^-\to\gamma \chi_{c2}$ channel, one more
additional resonance is added to describe the cross section line shape. Its
mass and width are measured to be $M=4371.7\pm7.5\pm1.8$ MeV/$c^2$ and
$\Gamma^{tot}=51.1\pm17.6\pm1.9$ 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 $e^+e^-\to\pi^+\pi^-\psi(3686)$, and the
$Y(4390)$ in $e^+e^-\to\pi^+\pi^-h_c$ within uncertainties. No significant
signal for the $e^+e^-\to\gamma \chi_{c0}$ process is observed, and the upper
limits of Born cross sections $\sigma_{B}(e^+e^-\to\gamma \chi_{c0})$ at 90\%
confidence level are reported.
