When I want to ask "what do you have for breakfast?", can I use any of the three, "che", "che cosa", or "cosa", to ask the question?
I'm not referring to specific structures like "che ore sono?".
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In the example sentence you can use any of them; but
Some purists frown upon cosa, but since they should blame Alessandro Manzoni for the usage, they can't say it too loudly. See http://www.mauriziopistone.it/testi/discussioni/gramm04_cosa.html for cosa as interrogative pronoun in Manzoni and other examples.
Addressing somebody with Che vuoi per colazione? may cause raised eyebrows in some parts of Italy and sound formal. Not too formal, though. Conversely, Cosa vuoi per colazione may sound funny in the Che regions.