  We update the study of the QCD corrections to direct J/psi and Upsilon
hadroproduction in association with a photon in the QCD-based approach of the
Colour-Singlet (CS) Model. After comparison with the recent full
next-to-leading-order (NLO) computation for this process, we provide an
independent confirmation to the inclusive case that NLO QCD corrections to
quarkonium-production processes whose LO exhibits a non-leading P_T behaviour
can be reliably computed at mid and large P_T by considering only the real
emission contributions accompanied with a kinematical cut. In turn, we evaluate
the leading part of the alpha^4_S alpha contributions, namely those coming from
(J/psi,Upsilon)+gamma associated with two light partons. We find that they are
dominant at mid and large P_T. This confirms our expectations from the leading
P_T scaling of the new topologies appearing at NNLO. We obtain that the yield
from the CS becomes one order of magnitude larger than the upper value of the
potential Colour-Octet yield. The polarization of the 3S1 quarkonia produced in
association with a photon is confirmed to be longitudinal at mid and large P_T.
