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