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We extend our original study in Ref. [1] on the production of $C=+$charmonium states $X=\eta_c(1S/2S)$ and $\chi_{cJ}(1P/2P)$ in$e^+e^-\to\gamma~+~X$ at B factories to the BEPCII/BESIII energy region with$\sqrt{s}=4.0\mbox{-}5.0$ GeV. In the framework of nonrelativistic QCDfactorization, the cross sections are estimated to be as large as$0.1\mbox{-}0.9$ pb. The results could be used to search for the missing $2P$charmonium states or to estimate the continuum backgrounds in the resonanceregion.

Search for $C=+$ charmonium states in $e^+e^-\to \gamma+~X$ at BEPCII/BESIII