Proton radiography has proved increasingly successful as a diagnostic for electric and magnetic fields in high-energy-density physics experiments. Most experiments use target-normal sheath acceleration sources with a wide energy range in the proton beam, since the velocity spread can help differentiate between electric and magnetic fields and provide time histories in a single shot. However, in magnetized plasma experiments with strong background fields, the broadband proton spectrum leads to velocity-spread-dependent displacement of the beam and significant blurring of the radiograph. We describe the origins of this blurring and show how it can be removed from experimental measurements, and we outline the conditions under which such deconvolutions are successful. As an example, we apply this method to a magnetized plasma experiment that used a background magnetic field of 3 T and in which the strong displacement and energy spread of the proton beam reduced the spatial resolution from tens of micrometers to a few millimeters. Application of the deconvolution procedure accurately recovers radiographs with resolutions better than 100 µm, enabling the recovery of more accurate estimates of the path-integrated magnetic field. This work extends accurate proton radiography to a class of experiments with significant background magnetic fields, particularly those experiments with an applied external magnetic field.