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$χ_{c0} \, ω$ production in $e^+e^-$ annihilation through $ψ(4160)$

We argue that the recent BESIII data on the cross section for the process $e^+e^- \to χ_{c0} \, ω$ in the center of mass energy range 4.21 - 4.42 GeV can be described by the contribution of the known charmonium-like resonance $ψ(4160)$ with the mass of about 4190\,MeV. The value of the coupling in the transition $ψ(4160) \to χ_{c0} \, ω$ needed for this mechanism is comparable to that in another known similar transition $χ_{c0}(2P) \to J/ψ\, ω$. The suggested mechanism also naturally explains the reported relative small value of the cross section for the final states $χ_{c1} \, ω$ and $χ_{c2} \, ω$ above their respective thresholds.