Based on your description of the cover, this is probably The Strayed Sheep of Charun (1977) by John Maddox Roberts. I originally thought it might be Cestus Dei (1983) which is an expanded re-write of the other, but the cover of the original fits your description better.
The rest of this is based on Cestus Dei because I can't find even a single review of The Strayed Sheep of Charun; nevertheless I'm guessing that much of the plot will be the same.
The entire settled galaxy is divided among the various major religions, that meet as a body called the "Unified Faiths" to mediate disputes and resolve problems.
The opening paragraph is:
Archbishop Hilarion occupied his rightful seat in the Great Hall of United Faiths with the stoicism of many years of experience. On the floor, a minor imam of some obscure Islamic sect was droning a speech welcoming the new representative from Shriva, which, despite its name, was a planet settled entirely by Mormons. The archbishop yawned expertly, without moving a muscle of his face.
In the story the newly re-discovered Flavian system is found to have a planet, Cadmus, creating genetically-engineered automata ("the soulless") which is an offense to all of the unified faiths. A Jesuit commando, Father Miles of Durga is sent by the Re-established Church of Rome to lead a mission try to convert the pagan population back to the Roman church. After receiving his orders he decides to take only the young Franciscan Friar Jeremiah who wishes to preach to the poor of the Flavian system in the Franciscan tradition.
This appears to be similar to the other section you recall:
"Now, young friar," said the priest, changing the subject with bewildering rapidity, "what will you be doing when we get among the Flavian worlds?"
"Why, Father," stammered the friar, "I hardly know. This is my first mission. I suppose I will have to begin proselytizing on my own, among the poor."
"On your own? In what by all reports is one of the most corrupt sectors we have ever come across? You were shocked by what you saw on Gravitas, and that was just a training school for priests."
"To be sure, Father, I am inexperienced, but I find that sophistication comes with travel."
"True enough." snorted the priest, "but your order doesn't give you sufficient preparation."
"I shall pray for you too. Friar. No wonder the casualty figures are so high among the Franciscans. Mark this, my boy: A ruling class that lives by exploitation of the less fortunate lives in perpetual terror. If they think that there is new thought stirring among those masses, then they will react with hysterical brutality."
There is more detail in this review on Goodreads.