The Kirk Party were a radical Presbyterian faction of the Scottish Covenanters during the Wars of the Three Kingdoms. They came to the fore after the defeat of the Engagers faction in 1648 at the hands of Oliver Cromwell and the English Parliament. They purged the Covenanters' General Assembly and army of "ungodly elements" and crowned Charles II as King of Scotland in 1651, in return for his explicit endorsement of their religious and political agenda in the Treaty of Breda (1650).
Kirk is a Scottish word meaning a church, or more specifically, the Church of Scotland.
The Kirk party's religious zeal did not help their cause militarily. In the month before the Battle of Dunbar they chose to institute a searching three-day examination of the political and religious sentiments of the Scottish army. The result was that the army was purged of "Malignants", 80 officers and 3000 experienced soldiers, while it lay within musket shot of the enemy.