The cross sections for inelastic photoproduction of J/psi, and psi' 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 psi' to J/psi 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 p(T). 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/psi cross sections were measured for 0.1 < z < 0.9, 60 < W < 240 GeV and p(T) > 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.