We have characterized the photodetection capabilities of single GaN nanowires incorporating 20 periods of AlN/GaN:Ge axial heterostructures enveloped in an AlN shell. Transmission electron microscopy confirms the absence of an additional GaN shell around the heterostructures. In the absence of a surface conduction channel, the incorporation of the heterostructure leads to a decrease of the dark current and an increase of the photosensitivity. A significant dispersion in the magnitude of dark currents for different single nanowires is attributed to the coalescence of nanowires with displaced nanodisks, reducing the effective length of the heterostructure. A larger number of active nanodisks and AlN barriers in the current path results in lower dark current and higher photosensitivity, and improves the sensitivity of the nanowire to variations in the illumination intensity (improved linearity). Additionally, we observe a persistence of the photocurrent, which is attributed to a change of the resistance of the overall structure, particularly the GaN stem and cap sections. In consequence, the time response is rather independent of the dark current. ; Comment: This document is the unedited Author's version of a Submitted Work that was subsequently accepted for publication in Nano Letters (2016), copyright (C) American Chemical Society after peer review. To access the final edited and published work see http://dx.doi.org/10.1021/acs.nanolett.6b00806