Based on accurate calculations of the flux of equivalent photons of the proton and heavy nuclei and the pQCD framework for the gluon distribution in the proton and nuclei, we analyze the rapidity and momentum transfer distributions of coherent $J/psi$ photoproduction in ultraperipheral proton-Pb collisions at the LHC. We demonstrate that unlike the case of proton-proton UPCs marred by certain theoretical uncertainties and experimental limitations, after a cut excluding the region of small momentum transfers, ultraperipheral proton-Pb collisions offer a clean way to study the gluon distribution in the proton down to $x approx 10^{-5}$. Our analysis of the momentum transfer distributions shows that an interplay of $J/psi$ production by low-energy photons on the nucleus and by high-energy photons on the proton in proton-Pb UPCs can result in some excess of events at small $p_t$ in a definite region of the rapidity y.
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