I mean, it was under the control of the monarch, but used canon law, and was lead by a senior cleric, typically an archbishop or cardinal.

If it helps, though, the Roman Inquisition, an organ of the Church, also liked to burn the odd apostate.

What Canon law are you referring to? What exactly was led by a senior cleric?

Frankly, it sounds as though you are randomly spouting phrases in an attempt to sound educated.