We present new results of a study of the submillimetre continuum emission from a sample of nine radio galaxies and four radio-quiet quasars at redshiftsz = 0.75-4.26. The observations were made at 800 μ m, using the single-element bolometer UKT14 on the James Clerk Maxwell Telescope (JCMT), reaching a typical rms sensitivity of σrms∼4 mJy, and they represent some of the deepest submillimetre extragalactic measurements made to date. Three detections were achieved, of which two are secure (4C 41.17 and H1413 + 117) and one (53W002) is tentative, whilst comparable upper limits were obtained for seven of the 10 remaining sources. We use these data as the motivation for a detailed discussion of the conversion from submillimetre and millimetre continuum fluxes to dust/gas masses and star formation rates at high redshift, and determine these quantities from our own and other data on high-redshift radio galaxies and quasars. In particular, we investigate the impact of the four main sources of uncertainty in deriving physical quantities from such data, namely (i) potential contamination by Galactic cirrus, (ii) uncertainty in the value of the dust rest-frequency mass-absorption coefficient, (iii) difficulty in constraining the dust temperature, and (iv) estimation of the appropriate gas-to-dust ratio in high-redshift objects. Our discussion emphasizes how important it will be to quantify and, where possible, minimize such uncertainties (via, for example, appropriate observational strategies) in order to capitalize fully on the 10-fold improvement in sensitivity offered by the imminent arrival of the next generation of bolometer arrays, such as SCUBA on the JCMT. Taking these uncertainties into account, we show that whilst the high-redshift galaxies discussed in this paper are amongst the most dust-rich and luminous objects discovered to date, their far-infrared properties are more comparable with those of the most luminous nearby interacting galaxies than with those expected of primaeval giant ellipticals. This conclusion is rather insensitive to the adopted dust temperature, and the appropriateness of our adopted gas-to-dust ratio is supported by the molecular line detections of lensed objects. Indeed, despite all of the uncertainties peculiar to studying galaxy evolution at submillimetre wavelengths, the current uncertainty over the true value of Ω0 is probably the most important factor affecting our conclusions.