I'm not Catholic so maybe I'm not authorized to answer this, but historically the Pope was considered “the most important person in the world” - at least the Western, Christian world. So in order to get an audience with the Pope you had to be extremely important yourself, and there was incredible protocol involved, even for Kings and Queens. The phrase was a common one that I recall hearing myself used in vernacular conversation. It was a put down used to express Jack’s frustration with Ennis.
Of course, Jacks family (wife) owned their business so it was easy for him to get vacation time off. Ennis who was still doing basic manual labor found it much more difficult and was at the mercy of his boss. As he say’s he “used to quit the jobs” in order to get time off to meet Jack, as he got older this option presumably became less viable if he wanted to move up in a position of responsibility and greater pay. If he didn't (or couldn't) make the required child support payments they'd likely take his truck, or worse, his horses.