In the Gospel of Mark, there is an account of Jesus performing an exorcism on a child. While Jesus was traveling around Caesarea Philippi, he reveals to his Apostles that he is the Messiah and that he will die and rise again after three days. Escaping the gathering crowd, Jesus took Peter, James and John to a mountain. There he “transfigured” or revealed his full majestic form in communion with Elijah and Moses. While Jesus was on the mountain he left the other nine Apostles behind with the crowd.
During this time, a man brought his demon-possessed son was brought to the Apostles, having heard of the miraculous healing powers these men possessed. The Apostles, while commissioned with special power by Jesus Christ, were not able to remove this demon. This led to an ensuing argument over whether Jesus and his Apostles really had the power to cast out demons and heal ailments. (Mark 8:27-9:14).
When Jesus returned, he saw the argument. The father went up to Jesus and asked him to heal his son- if he was able. On this Jesus questioned the father’s doubt- if he could heal. Jesus told to him that anything was possible for those who believe and the father cried “I believe, help me with my unbelief!”. So Jesus rebuked the demon and it left the boy. After leaving the presence of the crowd the Apostles asked Jesus why they weren’t able to drive out the demon. This sort of thing could not be resolved except by prayer (Mark 9:29) (some textual variants add fasting, but the point is the same).
“But if you can do anything, have compassion on us and help us. And Jesus said to [the father], ‘ “If you can”! All things are possible for one who believes.’ Immediately the father of the child cried out and said, ‘I believe; help my unbelief!’" (Mark 9:22b-24)
“This kind cannot be driven out by anything but prayer [and fasting]” (Mark 9:29)
"Truly I tell you, if you have faith as small as a mustard seed, you can say to this mountain, ‘Move from here to there,’ and it will move. Nothing will be impossible for you.” (Matt 17:20)
In this account there are a doubting father and self-reliant Apostles. The Apostles had the special authority of God to drive out demons (Mark 3:15). The father came to the Apostles, knowing that the Apostles had performed exorcisms before. When Jesus himself came he had only a sliver of hope left. He thought to himself “Jesus fix this if you can”. Jesus told him to believe and the father knew he needed help with unbelief. The Apostles could not drive it out themselves, because only prayer could drive it out. The Apostles had been given power by God, but weren’t relying on God to do the healing. Instead, they needed to be fully reliant on God through prayer (and perhaps fasting). The father did not get what he desired, because he did not truly believe the Apostles could do it.
The same could be said for believers today. We do not have the same supernatural powers that the Apostles did. Even still, God grants believers a spiritual power to deal with problems of this life and to wage a spiritual war (Eph 6:12). Despite this, there are still some things we cannot “cast away”. In many cases, the only thing we can do is pray and believe.
So whatever “irremovable demon” is in your life- remember that anything is possible with belief (Mark 9:23). We have power from God to live life, but sometimes the only thing we can do is let God do his work. As Christians, God has given us power to rely on his power, not to muster it up ourselves.