Do prophets make mistakes?

Yes, and the Restoration has said so about itself from its first printed page. The Book of Mormon opens by allowing that "if there are faults they are the mistakes of men." The Doctrine and Covenants says the Lord speaks to his servants "in their weakness, after the manner of their language."

Underneath the question is usually a worry: if a prophet can be wrong, how can I trust any of it? That worry assumes that what holds the Church together is the errorlessness of its leaders. What binds this people is a covenant: baptism, the sacrament table, the temple, a priesthood order, a life lived together. Prophets administer that covenant; they were never required to be flawless to administer it faithfully.

The books on the right go deeper than any page here can — each one opens in the archive at the exact page. Or read the full answer, with the hard case included.