Hyperluminous infrared galaxies (HyLIRGs) are the most extreme star-forming systems observed in the early Universe, and their properties still elude comprehensive understanding. We have undertaken a large XMM–Newton observing programme to probe the total accreting black hole population in three HyLIRGs at z = 2.12, 3.25, and 3.55, gravitationally lensed by foreground galaxies. Selected from the Planck All-Sky Survey to Analyse Gravitationally lensed Extreme Starbursts (PASSAGES), these HyLIRGs have apparent infrared luminosities >1014 L⊙. Our observations revealed X-ray emission in each of them. PJ1336+49 appears to be dominated by high-mass X-ray binaries (HMXBs). Remarkably, the luminosity of this non-AGN X-ray emission exceeds by a factor of about 3 the value obtained by calibration with local galaxies with much lower star formation rates. This enhanced X-ray emission most likely highlights the efficacy of dynamical HMXB production within compact clusters, which is an important mode of star formation in HyLIRGs. The remaining two (PJ0116−24 and PJ1053+60) morphologically and spectrally exhibit a compact X-ray component in addition to the extended non-AGN X-ray emission, indicating the presence of Active Galactic Nuclei (AGNs). The AGN appears to be centrally located in the reconstructed source plane images of PJ0116−24, which manifests its star-forming activity predominantly within an extended galactic disc. In contrast, the AGN in the field of PJ1053+60 is projected 60 kpc away from the extreme star-forming galaxy and could be ejected from it. These results underline the synergistic potential of deep X-ray observations with strong lensing for the study of high-energy astrophysical phenomena in HyLIRGs.