The far-IR/sub-mm wavelength range contains a wealth of diagnostic information that is important for understanding the role of active galactic nuclei (AGN) in galaxy evolution. Here we present the results of Herschel PACS and SPIRE observations of a complete sample of 46 powerful 2 Jy radio AGN at intermediate redshifts (0.05 < z < 0.7), which represent the deepest pointed observations of a major sample of radio AGN undertaken by Herschel. In order to assess the importance of non-thermal synchrotron emission at far-IR wavelengths, we also present new APEX sub-mm and ALMA mm data. We find that the overall incidence of non-thermal contamination in the PACS bands (<200 μm) is in the range 28-43 per cent; however, this rises to 30-72 per cent for wavelengths (>200 μm) sampled by the SPIRE instrument. Non-thermal contamination is strongest in objects with compact CSS/GPS or extended FRI radio morphologies, and in those with type 1 optical spectra. Considering thermal dust emission, we find strong correlations between the 100 and 160 μm monochromatic luminosities and AGN power indicators, providing further evidence that radiation from the AGN may be an important heating source for the far-IR emitting dust. Clearly, AGN contamination - whether by the direct emission from synchrotron-emitting lobes and cores, or via radiative heating of the cool dust - needs to be carefully considered when using the far-IR continuum to measure the star formation rates in the host galaxies of radio AGN.