\begin{document}$ J/\psi $\end{document}, \begin{document}$ \psi(2S) $\end{document}) and bottomonium (\begin{document}$ \Upsilon(nS) $\end{document}) photoproduction and fragmentation processes in p-p and \begin{document}$ \rm Pb$\end{document}-\begin{document}$\rm Pb$\end{document} collisions at LHC energies, where the ultra-incoherent photon emission is included. Using the NRQCD factorization approach, an exact treatment is developed. This approach recovers the Weizsäcker-Williams approximation (WWA) near the \begin{document}$ Q^{2}\sim0 $\end{document} region, where the Martin-Ryskin and BCCKL methods are used to avoid double counting. We calculated the \begin{document}$ Q^{2} $\end{document}, y, z, \begin{document}$ \sqrt{s} $\end{document}, and \begin{document}$ p_{T} $\end{document} dependent and total cross sections. Inelastic photoproduction and fragmentation were observed to contribute to heavy quarkonium production, particularly at large \begin{document}$ p_{T} $\end{document}. In addition, the contribution of the ultra-incoherent photon channel, which increases rapidly with quarkonium mass, is significant and begins to dominate the photoproduction processes for large \begin{document}$ p_{T} $\end{document}. We also obtained the complete WWA validity scopes of inelastic heavy quarkonium photoproduction in heavy-ion collisions. The WWA had high accuracy at high energies and for \begin{document}$\rm Pb $\end{document}-\begin{document}$\rm Pb $\end{document} collisions. However, current photon spectra are derived beyond the WWA scope, and double counting can occur when considering different channels simultaneously."> Inelastic heavy quarkonium photoproduction in <i>p</i>-<i>p</i> and Pb-Pb collisions at LHC energies -
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