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The $J/\psi$ production originating from elastic and inelasticphotoproduction processes in p-p collisions at LHC energies is investigated,where the fragmentation processes are involved. An exact treatment isperformed, which adopts the Martin-Ryskin method to weight the contributionfrom different channels, and can return to Weizs\"{a}cker-Williamsapproximation (WWA) when $Q^{2}\rightarrow0$. The relevant kinematicalrelations are also achieved. We present a comprehensive analysis for thefeature of WWA by comparing with the exact treatment. The results are expressedin $Q^{2}$, $y$, $z$, $p_{T}$, and $y_{r}$ (rapidity) distributions, and thetotal cross sections are also estimated. The numerical results indicate that,the incoherent-photon emission can provide the meaningful contribution toelastic photoproduction, and starts to play a very important role in theinelastic processes. The photoproduction and fragmentation processes canimprove the contribution of $J/\psi$ production in $p$-$p$ collisions at LHCenergies. Moreover, the WWA is only effective in very restricted domains, andthe exact treatment is needed to deal accurately with the $J/\psi$photoproduction, which can naturally avoid double counting and WWA errors.

Elastic and inelastic $J/\psi$ photoproduction in $p$-$p$ collisions at LHC energies: the feature of Weizs\"{a}cker-Williams approximation