That is important for musicians, but not technically a "prize" and those items are not included on this list.
This list includes competitions that are, generally speaking, open to "all-comers," although some of them impose age limits or residency (in Italy) requirements.
The list, below, divided by area of interest; that is, classical, jazz, opera, instrumental, etc.
Where possible, the competition name is linked to an external site, or an email address is provided where further details should be available.
Puglia International Piano Competition 'Mauro Paolo Monopoli Prize' Website 'Città di Barletta' Young Musician Intl.