  We study the process $\bar p p \rightarrow J/\psi + \gamma +X$ at the
Tevatron energy ($\sqrt{s} = 1.8$~TeV). The perturbative QCD contributions to
this process from the gluon-fusion and the fragmentation mechanisms are
computed. For the entire range of $p_T$ that can be probed at the Tevatron, the
fusion contribution is found to be dominant. Consequently the QCD prediction
for this process has better precision than the purely hadronic production of
$J/\psi$. An experimental study of this process at the Tevatron will help to
clarify several aspects of quarkonium production at large $p_T$.
