Perovskite solar cells (PSCs) represent a promising technology for highly efficient sunlight harvesting and its conversion to electricity at convenient costs. However, a few flaws of current devices undermine the long-term stability of PSCs. Some of them concern the interface between the photoactive perovskite and the hole transport layer (HTL), e.g. undesired charge recombination, polarization barriers and oxidation processes. A strategy to solve this problem is to replace the standard organic HTL (e.g. Spiro-OMeTAD) with a solid-state inorganic layer. Being extensively used in p-type dye sensitized solar cells (DSSCs), nickel oxide (NiO) has been the first choice as an inorganic HTL. Despite the great interests in the application of NiO and other p-type oxides in PSCs, there is no available atomistic model of their interface with a halide perovskite. Here, we address this knowledge gap via a thorough first-principles study of the prototypical PSC perovskite methyl-ammonium lead iodide (MAPI) and two inorganic p-type oxides: NiO and CuGaO2. This copper-gallium delafossite oxide is one of the most promising alternatives to NiO in p-type DSSCs, thanks to its wide optical bandgap and low valence band edge. Here, we characterize the properties of both isolated surface slabs and MAPI/HTL heterostructure models. Besides considering MAPI/NiO and MAPI/CuGaO2 interfaces from the pristine materials, we also address the effects of intrinsic and extrinsic p-type defects in both NiO (Ni vacancy, Ni vacancy with Li and Ag doping) and CuGaO2 (Cu vacancy) using more realistic models. Our study reveals the most convenient interfaces in terms of structural affinities and adhesion energies. From the electronic perspective, we present a detailed analysis on band edge alignments, with direct insights into the key functional parameters of PSCs: hole injection driving force and open circuit potential. Our data show how the presence of defects/dopants is crucial for a convenient hole injection in NiO and CuGaO2. These results provide new science-based design principles for further development of p-type oxides in PSC devices.