Context. Galaxy clusters in the local universe descend from high-redshift overdense regions known as protoclusters. The large gas reservoirs and high rate of galaxy interaction in protoclusters are expected to enhance star-formation activity and trigger luminous supermassive black-hole accretion in the nuclear regions of the host galaxies.Aims. We investigated the active galactic nucleus (AGN) content of a gas-rich and starbursting protocluster at z=4.002, known as the Distant Red Core (DRC). In particular, we search for luminous and possibly obscured AGN in 13 identified members of the structure, and compare the results with protoclusters at lower redshifts. We also test whether a hidden AGN can power the Ly alpha blob (LAB) detected with VLT/MUSE in the DRC.Methods. We observed all of the identified members of the structure with 139 ks of Chandra ACIS-S imaging. Being less affected by absorption than optical and IR bands, even in the presence of large column densities of obscuring material, X-ray observations are the best tools to detect ongoing nuclear activity in the DRC galaxies.Results. We detect obscured X-ray emission from the two most gas-rich members of the DRC, named DRC-1 and DRC-2. Both of them are resolved into multiple interacting clumps in high-resolution Atacama Large Millimeter Array and Hubble Space Telescope observations. In particular, DRC-2 is found to host a luminous (L2-10 keV approximate to 3x10(45) erg s(-1) ) Compton-thick (N-H greater than or similar to 10(24) cm(-2)) quasar (QSO) candidate, comparable to the most luminous QSOs known at all cosmic times. The AGN fraction among DRC members is consistent with results found for lower redshift protoclusters. However, X-ray stacking analysis reveals that supermassive black hole (SMBH) accretion is likely also taking place in other DRC galaxies that are not detected individually by Chandra.Conclusions. The luminous AGN detected in the most gas-rich galaxies in the DRC and the widespread SMBH accretion in the other members, which is suggested by stacking analysis, point toward the presence of a strong link between large gas reservoirs, galaxy interactions, and luminous and obscured nuclear activity in protocluster members. The powerful and obscured QSO detected in DRC-2 is likely powering the nearby LAB detected with VLT/MUSE, possibly through photoionization; however, we propose that the diffuse Ly alpha emission may be due to gas shocked by a massive outflow launched by DRC-2 over a approximate to 10 kpc scale.