  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, $x_\gamma^{\mathrm{meas}}$,
contributing to the photon-jet final state. Cross sections are evaluated in the
photon transverse-energy and pseudorapidity ranges $6 < E_T^{\gamma} < 15$ GeV
and $-0.7 < \eta^{\gamma} < 0.9$, and for jet transverse-energy and
pseudorapidity ranges $4 < E_T^{\rm jet} < 35$ GeV and $-1.5 < \eta^{\rm jet} <
1.8$, 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.
