An apparatus is provided for a turbine engine. This apparatus includes a fuel conduit and a fuel nozzle. The fuel conduit includes a supply passage. The fuel nozzle includes a nozzle passage, an end wall and a nozzle orifice. The nozzle passage has a longitudinal centerline and extends longitudinally through the fuel nozzle along the longitudinal centerline from the end wall to the nozzle orifice. The nozzle passage is configured with a convergent portion and a throat portion. The nozzle passage converges radially inward towards the longitudinal centerline as the convergent portion extends longitudinally along the longitudinal centerline away from the end wall and towards the throat portion. The supply passage is fluidly coupled to the nozzle passage by a fuel aperture in the end wall. A centerline of the fuel aperture is angularly and laterally offset from the longitudinal centerline.