  We calculate the cross section of J/psi plus prompt-photon inclusive
production in gamma gamma collisions at next-to-leading order within the
factorization formalism of nonrelativistic quantum chromodynamics (NRQCD)
focusing on direct photoproduction. Apart from direct J/psi production, we also
include the feed-down from directly-produced chi_{cJ} and psi' mesons. We
discuss the analytical calculation, in particular the treatment of the various
types of singularities and the NRQCD operator renormalization, in some detail.
We present theoretical predictions for the future e^+e^- linear collider TESLA,
taking into account both brems- and beamstrahlung.
