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As stated on the official Polizia di Stato website:
Foreigners awaiting renewal of their residence permits can leave and re-enter Italy if they hold:
The receipt issued by Italian Post offices (Poste Italiane S.p.A) certifying the submission of the application for renewal of their residence permit or EC residence permit for long-term residents;
The expired residence permit;
Their passport or other equivalent travel document;
they do not visit any other Schengen state.
So therefore, it is possible to leave the country and re-enter again with your receipt of renewal, as long as you follow the rules outlined above, but unfortunately you cannot go to Budapest as Hungary is a Schengen country.
Additionally, if you have not requested a renewal but your Residence Permit has expired less than 60 days ago, you can enter Italy as long as you are possession of a valid Visto di Reingresso, which is issued only in the presence of an authorized police station.