Game concept · By God Alone
Clerical Ruler Puppet
The game also links it as Clerical Ruler Puppets.
From the game's encyclopedia
A powerful member of the Clergy — an Archbishop or even the Head of Faith — that has become the Puppet of another Ruler.
The mightier the cleric, the greater the reach: a puppeted Head of Faith can Excommunicate your enemies, appoint Cardinals, grant a Pressed Claim, or incite a Holy War on your behalf.
The game's text for patch 1.20.1.2. Words in blue are entries of the encyclopedia: hover one to read it, as in the game.
Clerical Ruler Puppet is an entry of Crusader Kings III's in-game encyclopedia (By God Alone). The game explains it: "A powerful member of the Clergy — an Archbishop or even the Head of Faith — that has become the Puppet of another Ruler." It is part of Puppet. The game also links it as Clerical Ruler Puppets. Patch 1.20.1.2, from game/common/game_concepts/pam_game_concepts.txt and game/localization/english/game_concepts_l_english.yml.
Source: game/common/game_concepts/pam_game_concepts.txt, game/localization/english/game_concepts_l_english.yml and the English localisation, Crusader Kings III 1.20.1.2.