Approximately taking into account the higher-order effects due to multiple-gluon initial-state radiation, we extract from the latest Tevatron data of prompt J/ψ hadroproduction the leading colour-octet matrix elements within the nonrelativistic-QCD (NRQCD) factorization formalism proposed by Bodwin, Braaten, and Lepage. We find that the matrix elements which describe the formation of J/ψ mesons from colour-octet cc pairs in the angular-momentum states 2S+1 L J = 1 S 0 and 3 P J , with J = 0, 1, 2, which are responsible for the excess of the predicted cross section of inelastic J/ψ photoproduction over the existing HERA data at high values of the inelasticity variable z, are significantly reduced. We conclude that it is premature to proclaim a discrepancy between the Tevatron and HERA measurements of inclusive J/ψ production in the NRQCD framework. We also consider J/ψ mesons originating from the radiative feed down of promptly produced χ cJ mesons.