Can Prayer Change God's Mind?

In this episode Dr. Betters answers the question, "Hello! I am very much enjoying and benefiting from your Ministry, Dr Betters. Thank you. I have a question that follows on from your statement that prayer is never a way of trying to persuade God to change his mind. Given that there are Biblical examples of God being persuaded to change his mind and act differently to what He had first intended, why shouldn't people today plead humbly for God to change the course of events through prayer? (Abraham bargained with God over Sodom & Gomorrah, Hezekiah asked for extra years to live, the persistent mother in Matthew 15)."