Magnetoelectric multiferroics, either single-phase or composites comprising ferroelectric/ferromagnetic coupled films, are promising candidates for energy efficient memory computing. However, most of the multiferroic magnetoelectric systems studied so far are based on materials that are not compatible with industrial processes. Doped hafnia is emerging as one of the few CMOS-compatible ferroelectric materials. Thus, it is highly relevant to study the integration of ferroelectric hafnia into multiferroic systems. In particular, ferroelectricity in hafnia, and the eventual magnetoelectric coupling when ferromagnetic layers are grown atop of it, are very much dependent on quality of interfaces. Since magnetic metals frequently exhibit noticeable reactivity when grown onto oxides, it is expected that ferroelectricity and magnetoelectricity might be reduced in multiferroic hafnia-based structures. In this article, we present excellent ferroelectric endurance and retention in epitaxial Hf0.5Zr0.5O2 films grown on buffered silicon using Co as the top electrode. The crucial influence of a thin Pt capping layer grown on top of Co on the ferroelectric functional characteristics is revealed by contrasting the utilization of Pt-capped Co, non-capped Co and Pt. Magnetic control of the imprint electric field (up to 40% modulation) is achieved in Pt-capped Co/Hf0.5Zr0.5O2 structures, although this does not lead to appreciable tuning of the ferroelectric polarization, as a result of its high stability. Computation of piezoelectric and flexoelectric strain-mediated mechanisms of the observed magnetoelectric coupling reveal that flexoelectric contributions are likely to be at the origin of the large imprint electric field variation.