In SHELL/, _mandatory headers are typical header files of the form *_mandatory.h. They are generally included at the root of a group of API folders, and they contain the prototype for generic functions used by other files and systems throughout the engine.
In many cases, these functions are made mandatory in order to keep otherwise simple libraries cross-platform.
Perhaps the most relatable example of this is
show_error(). By having the function declared in a _mandatory header, it can be used by any system in ENIGMA on the premise that whatever the Widget API, something will implement it.
Any non-cross-platform function that is useful to cross-platform systems in ENIGMA is likely declared in a mandatory header.