In this extended analysis using the ZEUS detector at HERA, the photoproduction of isolated photons together with a jet is measured for different ranges of the fractional photon energy, $x_��^{\mathrm{meas}}$, contributing to the photon-jet final state. Cross sections are evaluated in the photon transverse-energy and pseudorapidity ranges $6 < E_T^�� < 15$ GeV and $-0.7 < ��^�� < 0.9$, and for jet transverse-energy and pseudorapidity ranges $4 < E_T^{\rm jet} < 35$ GeV and $-1.5 < ��^{\rm jet} < 1.8$, for an integrated luminosity of 374 $\mathrm{pb}^{-1}$. The kinematic observables studied comprise the transverse energy and pseudorapidity of the photon and the jet, the azimuthal difference between them, the fraction of proton energy taking part in the interaction, and the difference between the pseudorapidities of the photon and the jet. Higher-order theoretical calculations are compared to the results.

26 pages, 7 figures. Updated with small corrections from journal referee and journal reference