We present a detailed, broad-band X-ray spectral analysis of the ultraluminous X-ray source (ULX) pulsar NGC 7793 P13, a known super-Eddington source, utilizing data from the XMM–Newton, NuSTAR and Chandra observatories. The broad-band XMM–Newton+NuSTAR spectrum of P13 is qualitatively similar to the rest of the ULX sample with broad-band coverage, suggesting that additional ULXs in the known population may host neutron star accretors. Through time-averaged, phase-resolved and multi-epoch studies, we find that two non-pulsed thermal blackbody components with temperatures ∼0.5 and 1.5 keV are required to fit the data below 10 keV, in addition to a third continuum component which extends to higher energies and is associated with the pulsed emission from the accretion column. The characteristic radii of the thermal components appear to be comparable, and are too large to be associated with the neutron star itself, so the need for two components likely indicates the accretion flow outside the magnetosphere is complex. We suggest a scenario in which the thick inner disc expected for super-Eddington accretion begins to form, but is terminated by the neutron star's magnetic field soon after its onset, implying a limit of B ≲ 6 × 1012 G for the dipolar component of the central neutron star's magnetic field. Evidence of similar termination of the disc in other sources may offer a further means of identifying additional neutron star ULXs. Finally, we examine the spectrum exhibited by P13 during one of its unusual ‘off’ states. These data require both a hard power-law component, suggesting residual accretion on to the neutron star, and emission from a thermal plasma, which we argue is likely associated with the P13 system.