The X-ray spectra of Galactic binary systems dominated by a quasi-thermal component (disc dominated or high/soft state) are well described by a standard Shakura-Sunyaev disc structure down to the last stable orbit around the black hole. This is not the case in the very high (or steep power-law) state, where the X-ray spectra show both a strong disc component and strong, steep tail to higher energies. We use two such spectra from the black hole transient XTE J1550-564 as specific examples of this state, where the power emitted in the tail is more than 50 per cent of the bolometric luminosity. The simultaneous ASCA and RXTE data show that these have disc spectra which are significantly lower in temperature than those seen from the same source at the same luminosity in the high/soft state. If these give a true picture of the disc, then either the disc emissivity has reduced, and/or the disc truncates above the last stable orbit. However, it is often assumed that the tail is produced by Compton scattering, in which case its shape in these spectra requires that the Comptonizing region is marginally optically thick (τ ∼ 2-3), and covers a large fraction of the inner disc. This will distort our view of the disc, especially of the hottest-temperature material. We build a theoretical model of a Comptonizing corona over an inner disc, and fit this to the data, but find that it still requires a large increase in inner disc radius for a standard disc emissivity. Instead, it seems more probable that the disc emissivity changes in the presence of the corona. We implement the specific inner disc-corona coupling model of Svensson & Zdziarski, in which some fraction f of the accretion power is dissipated in the corona, leaving only a fraction 1 - f to be dissipated in the optically thick disc. We show that this can explain the low-temperature/high-luminosity disc emission seen in the very high state with only a small increase in radius of the disc. While this inferred disc truncation is probably not significant, given the model uncertainties, it is consistent with the low-frequency quasi-periodic oscillation and gives continuity of properties with the low/hardstate spectra.
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