We analyse the properties of the high-ionisation C IVλ1549 broad emission line in connection with the X-ray emission of 30 bright, optically selected quasars at z ≃ 3.0−3.3 with pointed XMM-Newton observations, which were selected to test the suitability of active galactic nuclei as cosmological tools. In our previous work, we found that a large fraction (≈25%) of the quasars in this sample are X-ray under-luminous by factors of > 3−10. As absorbing columns of ≳10²³ cm⁻² can be safely ruled out, their weakness is most likely intrinsic. Here we explore possible correlations between the UV and X-ray features of these sources to investigate the origin of X-ray weakness with respect to X-ray-normal quasars at similar redshifts. We fit the UV spectra from the Sloan Digital Sky Survey of the quasars in our sample and analyse their C IV properties – for example equivalent width (EW) and line peak velocity (υ_peak) – as a function of the X-ray photon index and 2−10 keV flux. We confirm the statistically significant trends of C IVυ_peak and EW with UV luminosity at 2500 Å for both X-ray-weak and X-ray-normal quasars, as well as the correlation between X-ray weakness (parametrised through Δα_ox) and C IV EW. In contrast to some recent work, we do not observe any clear relation between the 2−10 keV luminosity and υ_peak. We find a statistically significant correlation between the hard X-ray flux and the integrated C IV flux for X-ray-normal quasars, which extends across more than three (two) decades in C IV (X-ray) luminosity, whilst X-ray-weak quasars deviate from the main trend by more than 0.5 dex. We argue that X-ray weakness might be interpreted in a starved X-ray corona picture associated with an ongoing disc-wind phase. If the wind is ejected in the vicinity of the black hole, the extreme-UV radiation that reaches the corona will be depleted, depriving the corona of seed photons and generating an X-ray-weak quasar. Nonetheless, at the largest UV luminosities (> 10⁴⁷ erg s⁻¹) there will still be an ample reservoir of ionising photons that can explain the ‘excess’ C IV emission observed in the X-ray-weak quasars with respect to normal sources of similar X-ray luminosities.