Can a Muslim do something he swore to abandon while prostrating in prayer, because he thought it was forbidden, but then it became clear to him that it was not?

An oath is not formed except by its specific wording, and speaking during prayer is not permissible; it invalidates the prayer. If what prompted the oath was the swearer's belief that what he swore about was forbidden, then he does not break his oath if this belief disappears and it becomes clear that the matter is permissible. This is called "the basis of the oath" (bisat al-yameen) or "the incitement to the oath" (al-muhayyij li'l-yameen), which is the reason prompting the oath. If the swearer did not have an intention, then the basis of the oath particularizes and restricts the oath, and it is considered an implicit intention. Intention affects the oath by particularizing and generalizing it, and the reason takes its place when intention is absent. This ruling applies to all oaths.