  The inelastic charmonium ($J/\psi$, $\psi(2S)$, $\eta_{c}$, $h_{c}$ and
$\chi_{cJ}$) and bottomonium ($\Upsilon(nS)$, $\eta_{b}$, $h_{b}$, and
$\chi_{bJ}$) photoproductions in ultrarelativistic heavy-ion collisions at LHC
energies are studied, where the fragmentation processes are included. Based on
the factorization formalism of non-relativistic QCD, an exact treatment is
developed, which can weight the contribution from different channels and
recovers the Weizs\"{a}cker-Williams approximation (WWA) near the region
$Q^{2}\sim0$. The relevant kinematical relations are also achieved. We present
a comprehensive analysis for the properties of WWA in heavy-ion collisions, and
discuss the contribution of inelastic photoproduction processes to the heavy
quarkonium production. The $Q^{2}$-, $y$-, $\sqrt{s}$-, and $p_{T}$-dependent
cross sections, and the total cross sections are estimated. It is shown that
the inelastic photoproduction and fragmentation processes can provide the
evident modification to the heavy quarkonium production in $p$-$p$ and
$Pb$-$Pb$ collisions at LHC energies, where the ultra-incoherent photon
emission plays a very important role. Moreover, the WWA is only effective in
very restricted domains, and the exact treatment can naturally avoid double
counting and WWA errors.
