We analyse four XMM-Newton observations of the neutron-star low-mass X-ray binary EXO 0748$-$676 in quiescence. We fit the spectra with an absorbed neutron-star atmosphere model, without the need for a high-energy (power-law) component; with a 95 per cent confidence the power-law contributes less than 1 per cent to the total flux of the source in $0.5-10.0$ keV. The fits show significant residuals at around 0.5 keV which can be explained by either a hot gas component around the neutron star or a moderately broad emission line from a residual accretion disc. The temperature of the neutron-star has decreased significantly compared to the previous observation, from 124 eV to 105 eV, with the cooling curve being consistent with either an exponential decay plus a constant or a (broken) power law. The best-fitting neutron-star mass and radius can be better constrained if we extend the fits down to the lowest possible energy available. For an assumed distance of 7.1 kpc, the best-fitting neutron-star mass and radius are $2.00_{-0.24}^{+0.07}~M_odot$ and $11.3_{-1.0}^{+1.3}$ km if we fit the spectrum over the $0.3-10$ keV range, but $1.50_{-1.0}^{+0.4}~M_odot$ and $12.2_{-3.6}^{+0.8}$ km if we restrict the fits to the $0.5-10$ keV range. We finally discuss the effect of the assumed distance to the source upon the best-fitting neutron-star mass and radius. As systematic uncertainties in the deduced mass and radius depending on the distance are much larger than the statistical errors, it would be disingenuous to take these results at face value.
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