  We present a study of $J/\psi + jet$ diffractive production in the direct
photon process at HERA based on the factorization theorem for lepton-induced
hard diffractive scattering and the factorization formalism of the
nonrelativistic QCD (NRQCD) for quarkonia production. Using the diffractive
gluon distribution function extracted from HERA data on diffractive deep
inelastic scattering and diffractive dijet photon production, we show that this
process can be studied at HERA with present integrated luminosity, and can give
valuable insights in the color-octet mechanism for heavy quarkonia production.
