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We study the inelastic charmonium ($J/\psi$, $\psi(2S)$) and bottomonium($\Upsilon(nS)$) photoproduction and fragmentation processes in $p$-$p$ and$Pb$-$Pb$ collisions at LHC energies, where the ultra-incoherent photonemission is included. In the framework of the NRQCD factorization approach, anexact treatment is developed which recovers Weizs\"{a}cker-Williamsapproximation (WWA) near the region $Q^{2}\sim0$, where the methods ofMartin-Ryskin and BCCKL are used to avoid double counting. We calculate the$Q^{2}$, $y$, $z$, $\sqrt{s}$, $p_{T}$ dependent and the total cross sections.It turns out that the inelastic photoproduction and fragmentation processesprovide valuable contributions to the heavy quarkonium production, especiallyin the large $p_{T}$ regions. While the relative contribution ofultra-incoherent photon channel is very important, which rapidly increasesalong with the growing quarkonium mass, and begins to dominate thephotoproduction processes at large $p_{T}$ ranges. Moreover, we obtain thecomplete validity scopes of WWA in inelastic heavy quarkonium photoproductionin heavy-ion collisions. WWA has a much higher accuracy at high energies and in$Pb$-$Pb$ collisions. The existing photon spectra are generally derived beyondthe applicable scopes of WWA, and the double counting exists when the differentchannels are considered simultaneously.

Inelastic heavy quarkonium photoproduction in $p$-$p$ and $Pb$-$Pb$ collisions at LHC energies