Forward photoproduction of $J/psi$ can be used to extract Generalized Parton Distributions(GPDs) of gluons. We analyze the process at twist-3 level and study relevant classifications of twist-3 gluon GPDs. At leading power or twist-2 level the produced $J/psi$ is transversely polarized. We find that at twist-3 the produced $J/psi$ is longitudinally polarized. Our study shows that in high energy limit the twist-3 amplitude is only suppressed by the inverse power of the heavy quark mass relatively to the twist-2 amplitude. This indicates that the power correction to the cross-section of unpolarized $J/psi$ can have a sizeable effect. We have also derived the amplitude of the production of $h_c$ at twist-3, but the result contains end-point singularities. The production of other quarkonia has been briefly discussed.
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