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We study inclusive $J/\psi$ photoproduction at NLO at large $P_T$ at HERA andthe EIC. Our computation includes NLO QCD leading-$P_T$ corrections, QEDcontributions via an off-shell photon as well as those from $J/\psi$+charmchannels. For the latter, we employ the variable-flavour-number scheme. Ourresults are found to agree with the latest HERA data by H1 and provide, for thefirst time, a reliable estimate of the EIC reach for such a measurement.Finally, we demonstrate the observability of $J/\psi$+charm production and thesensitivy to probe the non-perturbative charm content of the proton at high$x$, also known as intrinsic charm, at the EIC.

NLO inclusive $J/\psi$ photoproduction at large $P_T$ at HERA and the EIC