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You are correct that Heresy was an offence dealt with by the Church courts. However, the church courts could indeed sentence heretics to death.
If convicted, the offender was then handed over to the State for that sentence to be carried out.
However, this does not mean that the State ordered the death of the offender. They merely carried out the sentence. They also had no discretion in the matter. As noted in the paper A Brief History of Medieval Roman Canon Law in England, the situation was made clear by the English Canonist, William Lyndwood in 1430:
In other words, "heresy was in his view a spiritual crime, and it was for the Church, not for the State, to say what should be done with heretics. If a lay prince refused to execute the Church's law concerning this crime, he was to be excommunicated."
There is a detailed exploration of how heresy was dealt with by the church courts in medieval England in Ian Forrest's 2005 book, The Detection of Heresy in Late Medieval England.
So, to summarise, heretics were charged by the church courts, convicted by the church courts, and sentenced by the church courts.
Only then were those heretics handed over to the State for that sentence to be carried out.