We present new X-ray observations of luminous heavily dust-reddened quasars (HRQs) selected from infrared sky surveys. HRQs appear to be a dominant population at high redshifts and the highest luminosities, and may be associated with a transitional blowout phase of black hole and galaxy co-evolution models. Despite this, their high-energy properties have been poorly known. We use the overall sample of $10$ objects with XMM-Newton coverage to study the high-energy properties of HRQs at $left< L_{rm bol} right> = 10^{47.5}$ erg/s and $left< z right>= 2.5$. For the seven sources with strong X-ray detections, we perform spectral analyses. These find a median X-ray luminosity of $left< L_{rm 2-10,keV} right> = 10^{45.1}$ erg/s, comparable to the most powerful X-ray quasars known. The gas column densities are $N_{rm H}=(1$-$8)times 10^{22}$ cm$^{-2}$, in agreement with the amount of dust extinction observed. The dust to gas ratios are sub-Galactic, but are higher than found in local AGN. The intrinsic X-ray luminosities of HRQs are weak compared to the mid-infrared ($L_{rm 6mu m}$) and bolometric luminosities ($L_{rm bol}$), in agreement with findings for other luminous quasar samples. For instance, the X-ray to bolometric corrections range from $kappa_{rm bol}approx 50$-$3000$. The moderate absorption levels and accretion rates close to the Eddington limit ($left< lambda_{rm Edd} right>=1.06$) are in agreement with a quasar blowout phase. Indeed, we find that the HRQs lie in the forbidden region of the $N_{rm H}$-$lambda_{rm Edd}$ plane, and therefore that radiation pressure feedback on the dusty interstellar medium may be driving a phase of blowout that has been ongoing for a few $10^{5}$ years. The wider properties, including [OIII] narrow-line region kinematics, broadly agree with this interpretation.
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