What was Henry ll's argument with the Church?

I don't think there is any real evidence indicating that Becket hated Henry II. On the contrary, it looks much more like Becket liked Henry. The problem was not hatred, but the fact that Becket had been given a job with importance to the church, and part of that job included protecting the church and the interests of the church. When Henry sought to acquire those interests for himself, Becket was duty bound not to let him. Becket was stubborn, but he seems to have been acting quite honestly and dispassionately.

What was the argument between Henry and Becket?

Becket broke his promise and betrayed him and turned very holy and thought because he was the archbishop he didnt have to listen and do what Henry V11 said.

Henry wanted more monarch power, but Thomas Becket disagreed. This caused dispute so Henry II killed him.