My question is concerning prayer. Should a Muslim pray with his head covered or uncovered, and does it matter?

Covering the head is not a requirement for praying. Until quite recently, wearing some sort of head covering was part of culture and tradition. So, from culture it crept into creed and then into religion.

We know from history and the Ahādīth literature that the Prophet (sws) himself prayed bare headed at times, which shows that this was no requisite of the prayer.