A fuel delivery apparatus is provided for a gas turbine engine. This fuel delivery apparatus includes a fuel injector. The fuel injector includes a plurality of fuel passages, a guide passage, a plurality of air passages, a mixing cavity and a fuel injector outlet. The fuel passages extend along an axis to the guide passage. The fuel passages converge radially inwards towards the axis as the fuel passages spiral about the axis towards the guide passage. The guide passage turns radially outwards away from the axis as the guide passage extends from the fuel passages to the mixing cavity. The air passages converge radially inwards towards the axis as the air passages extend axially to the mixing cavity. The mixing cavity fluidly couples the guide passage and the air passages to the fuel injector outlet.