We present a measurement of the fraction of cluster galaxies hosting X-ray bright active galactic nuclei (AGN) as a function of clustercentric distance scaled in units of r₅₀₀. Our analysis employs high-quality Chandra X-ray and Subaru optical imaging for 42 massive X-ray-selected galaxy cluster fields spanning the redshift range 0.2 < z < 0.7. In total, our study involves 176 AGN with bright (R < 23) optical counterparts above a 0.5-8.0 keV flux limit of 10^-14 erg cm^-2 s^-1. When excluding central dominant galaxies from the calculation, we measure a cluster galaxy AGN fraction in the central regions of the clusters that is~3 times lower than the field value. This fraction increases with clustercentric distance before becoming consistent with the field at ~2.5r₅₀₀. Our data exhibit similar radial trends to those observed for star formation and optically selected AGN in cluster member galaxies, both of which are also suppressed near cluster centres to a comparable extent. These results strongly support the idea that X-ray AGN activity and strong star formation are linked through their common dependence on available reservoirs of cold gas.