That isn't true, the Pope has the same impact he'd have on any catholic in the world...
The Lateran Pacts finally settled the Roman Question, defining the relation between Italy and the Pope, who 'til then didn't recognize Italy and pretty much waited and looked for any foreign help to re-establish the papal state.
Later our constitution affirmed how laïcité is the supreme principle of the state, abolishing catholicism as the state religion...
All in all, it was a tactful way to deal with it, the Vatican wasn't "created" out of the blue, but was a far more complex historical reality...
That imo, the church has way too many benefits it's another matter, but the only political influence it might have is with money or if a certain politician would be such a devout fanatic...
But as I said, the latter case is non-existant, as first and foremost, what matter to italian politicians is money and maintaining their position :P
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