  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.
