The possibility to engineer nanowire heterostructures with large bandgap variations is particularly interesting for technologically important broadband photodetector applications. Here we report on a combined study of design, fabrication, and optoelectronic properties of infrared photodetectors comprising four million n(+)in(+) InP nanowires periodically ordered in arrays. The nanowires were grown by metalorganic vapor phase epitaxy on InP substrates, with either a single or 20 InAsP quantum discs embedded in the i-segment. By Zn compensation of the residual n-dopants in the i-segment, the room-temperature dark current is strongly suppressed to a level of pA/NW at 1 V bias. The low dark current is manifested in the spectrally resolved photocurrent measurements, which reveal strong photocurrent contributions from the InAsP quantum discs at room temperature with a threshold wavelength of about 2.0 m and a bias-tunable responsivity reaching 7 A/W@1.38 m at 2 V bias. Two different processing schemes were implemented to study the effects of radial self-gating in the nanowires induced by the nanowire/SiOx/ITO wrap-gate geometry. Summarized, our results show that properly designed axial InP/InAsP nanowire heterostructures are promising candidates for broadband photodetectors.