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Within the framework of the non-relativistic QCD (NRQCD), we make asystematical study of the yields and polarizations of $J/\psi$ and $\Upsilon$via $\gamma \gamma \to J/\psi(\Upsilon)+\gamma+X$ in photon-photon collisionsat the Circular Electron Positron Collider (CEPC), up to $\mathcalO(\alpha^{3}\alpha_s)$. We find that this process at CEPC is quite "clean",namely the direct photoproduction absolutely dominate over the single- anddouble- resolved processes, at least 2 orders of magnitude larger. It is foundthat the next-to-leading order (NLO) QCD corrections will significantly reducethe results due to that the virtual corrections to $^3S_1^1$ is large andnegative. For $J/\psi$, as $p_t$ increases, the color octet (CO) processes willprovide increasingly important contributions to the total NLO results. Moreoverthe inclusion of CO contributions will dramatically change the polarizations of$J/\psi$ from toally transverse to longitudinal, which can be regarded as adistinct signal for the CO mechanism. However, for the case of $\Upsilon$, theeffects of the CO processes are negligible, both for yields and polarizations.For $J/\psi$, the dependence of the yields on the value of the renormalizationscale $\mu_r$ is moderate, while significant for the polarization. The impactof the variation of $\mu_{\lambda}$ is found to be relatively slight. As forthe case of $\Upsilon$, the uncertainties of $\mu_{r}$ and $\mu_{\lambda}$ justbring about negligible effects. The future measurements on this semi-inclusivephotoproductions of $J/\psi(\Upsilon)+\gamma+X$, especially on the polarizationparameters of $J/\psi$, will be a good laboratory for the study of heavyquarkonium production mechanism and helpful to clarify the problems of the$J/\psi$ polarization puzzle.

$\mathcal O(\alpha^{3}\alpha_s)$ Study on the yields and polarizations  of $J/\psi(\Upsilon)$ within the framework of non-relativistic QCD via  $\gamma\gamma \to J/\psi(\Upsilon)+\gamma+X$ at CEPC