  We perform a survey of the X-ray properties of 41 objects from the WISE/SDSS
selected Hyper-luminous (WISSH) quasars sample, composed by 86 broad-line
quasars (QSOs) with bolometric luminosity $L_{Bol}\geq 2\times 10^{47}\,erg\,
s^{-1}$, at z~2-4. All but 3 QSOs show unabsorbed 2-10 keV luminosities
$L_{2-10}\geq10^{45} \,erg \,s^{-1}$. Thanks to their extreme radiative output
across the Mid-IR-to-X-ray range, WISSH QSOs offer the opportunity to
significantly extend and validate the existing relations involving $L_{2-10}$.
We study $L_{2-10}$ as a function of (i) X-ray-to-Optical (X/O) flux ratio,
(ii) mid-IR luminosity ($L_{MIR}$), (iii) $L_{Bol}$ as well as (iv)
$\alpha_{OX}$ vs. the 2500$\mathring{A}$ luminosity. We find that WISSH QSOs
show very low X/O(<0.1) compared to typical AGN values; $L_{2-10}/L_{MIR}$
ratios significantly smaller than those derived for AGN with lower luminosity;
large X-ray bolometric corrections $k_{\rm Bol,X}\sim$ 100-1000; and steep
$-2<\alpha_{OX}<-1.7$. These results lead to a scenario where the X-ray
emission of hyper-luminous quasars is relatively weaker compared to
lower-luminosity AGN. Models predict that such an X-ray weakness can be
relevant for the acceleration of powerful high-ionization emission line-driven
winds, commonly detected in the UV spectra of WISSH QSOs, which can in turn
perturb the X-ray corona and weaken its emission. Accordingly, hyper-luminous
QSOs represent the ideal laboratory to study the link between the AGN energy
output and wind acceleration. Additionally, WISSH QSOs show very large BH
masses ($\log[M_{\rm BH}/M_{\odot}]$>9.5). This enables a more robust modeling
of the $\Gamma-M_{BH}$ relation by increasing the statistics at high masses. We
derive a flatter $\Gamma$ dependence than previously found over the broad range
5 <$\log(M_{\rm BH}/M_{\odot})$ < 11.
