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 468 pb-1 collected in the period 1996-2007. The ψ′ to J/ψ cross section ratio was measured in the range 0.55 < z < 0.9 and 60 < W < 190 GeV as a function of W, z and pT. Here W denotes the photon-proton centre-of-mass energy, z is the fraction of the incident photon energy carried by the meson and p T is the transverse momentum of the meson with respect to the beam axis. The J/ψ cross sections were measured for 0.1 < z < 0.9, 60 < W < 240 GeV and pT > 1 GeV. Theoretical predictions within the non-relativistic QCD framework including NLO colour-singlet and colour-octet contributions were compared to the data, as were predictions based on the k T-factorisation approach. © 2013 SISSA.