Relativistic corrections to heavy quarkonium $+\gamma$ production at super Z factory

In the framework of nonrelativistic quantum chromodynamics, we study the $\mathcal{O}(v^2)$ relativistic corrections to heavy-quarkonium production associated with a photon at future electron-positron colliders operated in the Z-factory mode. S-, P-, and D-wave heavy quarkonia as well as the mixed state $\psi (3770)$ are considered. We comprehensively investigate tree-level $s$-channel processes for all states, together with $t$-channel contributions that apply only to ${}^3{L}_1$ states. It is found that the relativistic corrections substantially reduce the LO cross sections for charmonium states by roughly $20\mathrm{\%}$-$70\mathrm{\%}$, while the reduction is milder for bottomonium, at the level of $10\mathrm{\%}$-$30\mathrm{\%}$. Event yields for all studied channels are calculated, and sizable event yields are expected for ${\eta }_c$, $J\mathrm{/}\psi$, $\psi (2S)$, $\mathrm{{\rm Y}}$, and $\psi (3770)$, especially at FCC-ee.