  We revisit inclusive J/psi and Upsilon photoproduction at lepton-hadron
colliders, namely in the limit when the exchanged photon is quasi real. Our
computation includes the next-to-leading-order (NLO) alpha_s corrections to the
leading-order contributions in v. Similarly to the case of NLO
charmonium-hadroproduction processes, the resulting cross sections obtained in
the MS-bar factorisation scheme are sometimes found to be negative. We show
that the scale-fixing criteria which we derived in a previous study of eta(c)
production successfully solves this problem from the EicC all the way up to the
FCC-eh energies. We then elaborate on how to study a scale uncertainty akin to
that derived by scale variations when one fixes a scale. In turn, we
investigate where both J/psi and Upsilon photoproduction could be used to
improve our knowledge of gluon content of the proton at scales as low as a
couple of GeV.
