If you want the authentic, full-featured experience, you must use official hardware.
Any website promising a direct, official "download" of this specific game for the PSP is likely providing a or an unofficial homebrew project. The Reality of Mario Kart on PSP
A fan-made project developed for the PSP that features 10 characters and 10 stages. It is a standalone game, not a port of the Switch version.
Because Nintendo owns the Mario Kart franchise, their games never release on Sony hardware. However, the dedicated homebrew community has created ways to experience similar gameplay on the PSP:
While the search term is popular, it is important to clarify that Mario Kart 8 Deluxe is not officially available for the PlayStation Portable (PSP) . The game is an exclusive title for the Nintendo Switch .