One of the conditions for many indulgences is prayers for the intentions of the Holy Father. Few people know what this actually means. Often, this is misunderstood to be praying for the intentions of the current Pope when, in reality, this means something much more lasting and beautiful. The Church has defined the Pope’s intention as including four basic and universally Catholic intentions.
The Raccolta, which is a collection of indulgences, explains that this requirement for the gaining of indulgence states:
23. The Pope’s intention always includes the following objects:
i. The progress of the Faith and triumph of the Church.ii. Peace and union among Christian Princes and Rulers.iii. The conversion of sinners.iv. The uprooting of heresy,These intentions may be fittingly recalled and prayed for after Communion, or at the visit; but a general intention of praying according to the mind of the Pope is sufficient ; and any prayers may be used which are not already of obligation, e.g., the Little Hours of a priest’s Office. (July 12, 1847; May 29, 1841.)
Raccolta, xiii, 1910, http://staugustinechapel.org/Doctrine/Raccolta_1910.pdf
Let us all try to learn and understand the prayers we are saying! These intentions are quite beautiful and should help us pray more fervently instead of simply through habit.
God Bless
Another interesting reference on the subject from the SSPX in praying for the Pope:
https://sspx.org/en/pray-much-for-the-holy-father-popes-intentions-sspx
Vita Spiritualis Blog 