What are the flaws in the argument: “If you can change the will of God through prayer, it is subject to the power of prayer and therefore not all-powerful?”

The power you speak of is not in the human launching the prayer request. That is simply information. The power is in the Divine, who can answer the prayer in many, many differing ways, depending on all the variables of the equation. Does the person have sufficient belief in the Divine? Does the person believe in their own self-worth or have they given up and believe now it is hopeless, and they are helpless, and nothing can be done? We cannot, on the one hand, give human the power of free will and autonomy, and then dishonor their choice to not believe in themselves and in their own future, and override it with an agenda of our own. Even though that might be higher and better for them, in the long run, it would undercut them and short-circuit the learning needed to overcome such a dilemma. This they need to do largely on their own. We can help from behind the scenes, but we cannot always solve these problems quickly because the best solution may require that the person learn and grow their way to an answer and that is typically slow and painful and may require more than one lifetime. We have the patience to allow this to happen because, in the end, it will serve them far greater than a series of lifetimes where we rush in to compensate for mistakes and then the learning is less, and the growth is less, and they ultimately never achieve their true potential because they have become so dependent on us, they fail to fully develop their own capability.