  We study the effects of intrinsic transverse momentum smearing on J/psi
production both at the Tevatron and at HERA. For the case of large-p_T J/psi
production at the Tevatron, the effects due to k_T smearing are mild. On the
other hand, inelastic J/psi photoproduction at HERA is very sensitive to the
k_T smearing and, in fact, with a reasonable value of <k_T> it is possible to
resolve the large-z discrepancy seen by comparing non-relativistic QCD (NRQCD)
predictions with the HERA data. We conclude that, with the present kinematic
cuts, photoproduction at HERA is not a good test of NRQCD.
