Using data samples of 89.5~fb$^{-1}$, 711.0~fb$^{-1}$, and 121.4~fb$^{-1}$ collected with the Belle detector at the KEKB asymmetric-energy $e^+e^-$ collider at center-of-mass energies 10.52 GeV, 10.58 GeV, and 10.867 GeV, respectively, we study the exclusive reactions $e^+e^-togammachi_{cJ}$ ($J = 0,~1,~2$) and $e^+e^-togammaeta_c$. A significant $gamma chi_{c1}$ signal is observed for the first time at $sqrt{s}=10.58$ GeV with a significance of $5.1sigma$ including systematic uncertainties. No significant excesses for $gamma chi_{c0}$, $gamma chi_{c2}$, and $gamma eta_c$ final states are found, and we set 90% credibility level upper limits on the Born cross sections ($sigma_{rm B}$) at 10.52 GeV, 10.58 GeV, and 10.867~GeV. Together with cross sections measured by BESIII at lower center-of-mass energies, the energy dependency of $sigma_{rm B}(e^+e^-togammachi_{c1})$ is obtained.