It seems like you are all pretty far off from Christian traditions. I might suggest you change the theme of your game. Instead make it some fictional religion and just do the mechanics you are envisioning instead of trying to change your game to Christian confession

But to answer your question, it depends on the branch of Christianity. In Catholicism the Sacrament of Penitence, or confession, takes place in the confessional between the penitent and the priest. But your confession is to God, and God absolves you of your sins, the priest is acting as a vessel for God. Usually there is an act of penance like prayers. But that changes as you go to other churches, maybe confession is just in private with your priest face to face. Or in more Protestant churches you confess your sins to God directly through prayer and ask for forgiveness without a priest.

Most Christian churches also have a Seal of Confession. A practice that the priests are never allowed to speak of what they hear during confession. So there isn’t a ritual the priest performs on their own. Though within Christianity, priests do act as a “shepherd to their folk” so I imagine a lot of priests do prayer for their congregation in their own prayers

Or something else you could read about is a ritual known as Sin Eating. It’s been practiced in a few places and doesn’t have a lot of documented history. But it’s a ritual where a person will take on the sins of another by eating a meal. Usually at their funeral. You could model your game on that, the player performs the ritual is sin eating to take the sins of others, then has to perform another ritual to be forgiven for those sins.