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We consider the production of the $\eta_c$ and $\chi_{cJ}$ states recoiledwith a photon up to $\mathcal{O}(\alpha_s v^2)$ at BESIII and B-factorieswithin the frame of NRQCD factorization. With the corrections, we revisit thenumerical calculations to the cross sections for the $\eta_c(nS)$ and the$\chi_{cJ}(mP)$ states. We argue that the search for $XYZ$ states with evencharge conjugation such as $X(3872)$, $X(3940)$, $X(4160)$, and $X(4350)$recoiled with a photon at BESIII may help clarify the nature of these states.For completeness, the production of charmonium with even charge conjugationrecoiled with a photon at B factories is also discussed.

$\alpha_sv^2$ corrections to $\eta_c$ and $\chi_{cJ}$ production recoiled with a photon at $e^+e^-$ colliders