We study the inclusive J/psi production at large transverse momenta at lepton-hadron colliders in the limit when the exchange photon is quasi real, also referred to as photoproduction. Our computation includes the leading-P_T leading-v next-to-leading alpha_s corrections. In particular, we consider the contribution from J/psi plus another charm quark, by employing for the first time in quarkonium photoproduction the variable-flavour-number scheme. We also include a QED-induced contribution via an off-shell photon which remained ignored in the literature and which we show to be the leading contribution at high P_T within the reach of the EIC. In turn, we use our computation of J/psi+charm to demonstrate its observability at the future EIC and the EIC sensitivity to probe the non-perturbative charm content of the proton at high x.
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