Highly accreting active galactic nuclei (AGNs) have unique features that make them ideal laboratories for studying black hole accretion physics under extreme conditions. However, our understanding of the nuclear properties of these sources is hampered by the lack of a complete systematic investigation of this AGN class in terms of their main spectral and variability properties, and by the relative paucity of them in the local Universe, especially those powered by supermassive black holes with MBH > 108 M⊙. To overcome this limitation, we present here the X-ray spectral analysis of a new, large sample of 61 highly accreting AGNs named as the XMM-Newton High-Eddington Serendipitous AGN Sample, or X-HESS, obtained by cross-correlating the 11th release of the XMM-Newton serendipitous catalogue and the catalogue of spectral properties of quasars from the SDSS DR14. The X-HESS AGNs are spread across wide intervals with a redshift of 0.06 < z < 3.3, a black hole mass of 6.8 < log(MBH/M⊙) < 9.8, a bolometric luminosity of 44.7 < log(Lbol/erg s−1) < 48.3, and an Eddington ratio of −0.2 < log λEdd < 0.5, and more than one third of these AGNs can rely on multiple observations at different epochs, allowing us to complement their X-ray spectroscopic study with a variability analysis. We find a large scatter in the Γ − λEdd distribution of the highly accreting X-HESS AGNs, in agreement with previous findings. A significant correlation is only found by considering a sample of lower-λEdd AGNs with λEdd ≲ 0.3. We get hints that the Γ − λEdd relation appears to be more statistically sound for AGNs with lower MBH and/or Lbol. We investigate the possibility of transforming the Γ − λEdd plane into a fully epoch-dependent frame by calculating the Eddington ratio from the simultaneous optical/UV data from the optical monitor, λEdd, O/UV. Interestingly, we recover a significant correlation with Γ and a spread roughly comparable to that obtained when Lbol is estimated from SDSS spectra. Finally, we also get a mild indication of a possible anti-correlation between Γ and the strength of the soft excess, providing hints that reflection from an ionised disc may be effective in at least a fraction of the X-HESS AGNs, though Comptonisation from hot and warm coronae cannot be ruled out as well.