The photoproduction of prompt photons, together with an accompanying jet, has been studied in ep collisions at a centre-of-mass energy of 318 GeV with the ZEUS detector at HERA using an integrated luminosity of 77 pb(-1). Cross sections were measured for the transverse energy of the photon and the jet larger than 5 and 6 GeV, respectively. The differential gamma+jet cross sections were reconstructed as functions of the transverse energy, pseudorapidity and x(gamma) (obs), the fraction of the incoming photon momentum taken by the photon-jet system. Predictions based on leading-logarithm parton-shower Monte Carlo models and next-to-leading-order (NLO) QCD generally underestimate the cross sections for the transverse energies of prompt photons below 7 GeV, while the k(T)-factorisation QCD calculation agrees with the data better. When the minimum transverse energy of prompt photons is increased to 7 GeV, both NLO QCD and the k(T)-factorisation calculations are in good agreement with the data.