The cross sections for inelastic photoproduction of J/ψ and ψ′ mesons have been measured in ep collisions with the ZEUS detector at HERA, using an integrated luminosity of 38.0 pb-1. The events were required to have 0.1 \u3c z \u3c 0.9 and 50 \u3c W \u3c 180 GeV, where z is the fraction of the incident photon energy carried by the J/ψ in the proton rest frame and W is the photon-proton centre-of-mass energy. The ψ′ to J/ψ cross-section ratio was measured in the range 0.55 \u3c z \u3c 0.9. The J/ψ data, for various ranges of transverse momentum, are compared to theoretical models incorporating colour-singlet and colour-octet matrix elements. Predictions of a next-to-leading-order colour-singlet model give a good description of the data, although there is a large normalisation uncertainty. The J/ψ helicity distribution for z \u3e 0.4 is compared to leading-order QCD predictions
