How during the Middle Ages how did Catholic religious leaders influence the kings and lords of Europe?

Answer 1

The short answer is, a lot of different ways, chief among them are the following:

1. By placing political leaders under Interdict or Excommunication. Interdict means the person under interdict cannot receive the sacraments, excommunication means they not only cannot receive the sacraments, but are officially outside the Church. This would create unrest in the lord's domain, forcing action

2. Marriages. In the Catholic Church, there is no remarriage after divorce. The only way to get married if you have divorced your wife was to have the marriage declared null and void, declaring there never was a marriage. This is supposed to be done in cases where there was an impediment to marriage, say you married a woman who you later found out was your half sister, and this it is invalid due to consanguinity. Or lets say a woman marries a man who had no intention to be faithful, that would mean he never consented to the vows, thus the marriage is invalid. This was sometimes used as a political tool, grating an annulment in exchange for some political favor.

3. Politicing. Remember, the Papacy held land in the medieval times, so as a leader of a state, the Pope could influence other powers just as any other sovergn could.

Keep in mind, usually the influence went the other direction, kings and lords trying to influence the Church by appointing their own bishops and besieging the papal lands. Henry VIII asked for an annulment to Catherine of Aragon, which was impossible to grant due to Catherine's nephew being the Holy Roman Empire, which held significant influence in the Papal States.