I compute the leading color-octet contributions to the process gamma+p
-->J/psi+gamma(+X) within the nonrelativistic QCD (NRQCD) factorizatio
n formalism. In the color-singlet model, J/sigma+gamma can only be pro
duced when the photon interacts through its structure function, while
the color-octet mechanism allows for production of J/psi+gamma via dir
ect photon-gluon fusion. Resolved photon processes can be distinguishe
d easily from direct photon processes by examining the fraction of the
incident photon energy carried away by the J/psi in the event. Theref
ore, this process provides a conclusive test of the color-octet mechan
ism. J/psi+gamma production is particularly sensitive to the NRQCD mat
rix element which figures prominently in the fragmentation production
of J/psi at large p(perpendicular to) in hadron colliders. I also exam
ine the predictions of the color evaporation model (CEM) of quarkonium
production and find that this process can easily discriminate between
the NRQCD factorization formalism and the CEM.