jats:pThe hard X-ray emission in active galactic nuclei (AGN) and black hole X-ray binaries is thought to be produced by a hot cloud of electrons referred to as the corona. This emission, commonly described by a power law with a high-energy cutoff, is suggestive of Comptonization by thermal electrons. While several hypotheses have been proposed to explain the origin, geometry, and composition of the corona, we still lack a clear understanding of this fundamental component. jats:italicNuSTAR</jats:italic> has been playing a key role improving our knowledge of X-ray coronæ thanks to its unprecedented sensitivity above 10 keV. However, these constraints are limited to bright, nearby sources. The jats:italicHigh Energy X-ray Probe</jats:italic> (jats:italicHEX-P</jats:italic>) is a probe-class mission concept combining high spatial resolution X-ray imaging and broad spectral coverage (0.2–80 keV) with a sensitivity superior to current facilities. In this paper, we highlight the major role that jats:italicHEX-P</jats:italic> will play in further advancing our insights of X-ray coronæ notably in AGN. We demonstrate how jats:italicHEX-P</jats:italic> will measure key properties and track the temporal evolution of coronæ in unobscured AGN. This will allow us to determine their electron distribution and test the dominant emission mechanisms. Furthermore, we show how jats:italicHEX-P</jats:italic> will accurately estimate the coronal properties of obscured AGN in the local Universe, helping address fundamental questions about AGN unification. In addition, jats:italicHEX-P</jats:italic> will characterize coronæ in a large sample of luminous quasars at cosmological redshifts for the first time and track the evolution of coronæ in transient systems in real time. We also demonstrate how jats:italicHEX-P</jats:italic> will enable estimating the coronal geometry using spectral-timing techniques. jats:italicHEX-P</jats:italic> will thus be essential to understand the evolution and growth of black holes over a broad range of mass, distance, and luminosity, and will help uncover the black holes’ role in shaping the Universe.</jats:p>