Galaxies within galaxy clusters are known to be subject to a wide variety of environmental effects, both gravitational and hydrodynamical. In this study, we examine the purely gravitational interaction of idealized galaxy models falling into a galaxy cluster in the context of Modified Newtonian Dynamics (MOND). Thismodification of gravity gives rise to an external field effect (EFE), where the internal dynamics of a system are affected by the presence of external gravitational fields.We examine the consequences of the EFE on low- and high-mass disc galaxies in time-evolving analytic background cluster potentials, considering orbits with weak and strong tidal fields. By varying the orbital plane of the galaxies, we also test the effect of having the tidal interaction orthogonal or parallel to the disc. Furthermore, we consider as a control sample models where the EFE has been removed and they are only affected by tides. Our results suggest that MOND cluster galaxies should exhibit clear asymmetries in their isophotes, suffer increased mass loss and a reduction in their rotation curves due to the combined effect of cluster tides and the external field. In particular, low-mass galaxies are hit hard by the EFE, becoming dominated by dispersion rather than rotation even in the absence of tides.