[en] We have performed a systematic study of $J/\mathrm{\psi <!--\; ??\; -->}$ and $\mathrm{\psi <!--\; ??\; -->}(2S)$ production in p–p collisions at different Large Hadron Collider (LHC) energies and at different rapidities using the leading order non-relativistic quantum chromodynamics model of heavy quarkonium production. We have included the contributions from ${\mathrm{\chi <!--\; ??\; -->}}_{cJ}$ (J = 0, 1, 2) and $\mathrm{\psi <!--\; ??\; -->}(2S)$ decays to $J/\mathrm{\psi <!--\; ??\; -->}$. The calculated values have been compared with the available data from the four experiments at LHC, namely ALICE, ATLAS, CMS and LHCb. In the case of ALICE, inclusive $J/\mathrm{\psi <!--\; ??\; -->}$ and $\mathrm{\psi <!--\; ??\; -->}(2S)$ cross-sections have been calculated by including the feed-down from B mesons using a fixed-order next-to-leading logarithm formalism. It is found that all the experimental cross-sections are well described for $p_T><!--\; >\; -->$ 3 GeV within the theoretical uncertainties arising due to the choices of the factorization and renormalization scales. We also predict the transverse momentum distributions of $J/\mathrm{\psi <!--\; ??\; -->}$ and $\mathrm{\psi <!--\; ??\; -->}(2S)$ both for the direct and feed-down processes at the upcoming LHC energies of $\sqrt{s}=$ 5.1 and 13 TeV for the year 2015. (paper)