Observational constraints on the average radial distribution profile of AGN in distant galaxy clusters can provide important clues on the triggering mechanisms of AGN activity in dense environments and are essential for a completeness evaluation of cluster selection techniques in the X-ray and mm wavebands. The aim of this work is a statistical study with XMM-Newton of the presence and distribution of X-ray AGN in the large-scale structure environments of 22 X-ray luminous galaxy clusters in the redshift range 0.9 < z less than or similar to 1.6 compiled by the XMM-Newton Distant Cluster Project (XDCP). To this end, the X-ray point source lists from detections in the soft band (0.35-2.4 keV) and full band (0.3-7.5 keV) were stacked in cluster-centric coordinates and compared to average background number counts extracted from three independent control fields in the same observations. A significant full-band (soft-band) excess of similar to 78 (67) X-ray point sources is found in the cluster fields within an angular distance of 8' (4 Mpc) at a statistical confidence level of 4.0 sigma (4.2 sigma), corresponding to an average number of detected excess AGN per cluster environment of 3.5 +/- 0.9 (3.0 +/- 0.7). The data point towards a rising radial profile in the cluster region (r < 1Mpc) of predominantly low-luminosity AGN with an average detected excess of about one point source per system, with a tentative preferred occurrence along the main cluster elongation axis. A second statistically significant overdensity of brighter soft-band-detected AGN is found at cluster-centric distances of 4'-6' (2-3 Mpc), corresponding to about three times the average cluster radius R-200 of the systems. If confirmed, these results would support the idea of two different physical triggering mechanisms of X-ray AGN activity in dependence of the radially changing large-scale structure environment of the distant clusters. For high-z cluster studies at lower spatial resolution with the upcoming eROSITA all-sky X-ray survey, the results suggest that cluster-associated X-ray AGN may impose a bias in the spectral analysis of high-z systems, while their detection and flux measurements in the soft band may not be significantly affected.