Sure, but what is the idea behind modern Christianity's idea that prayer for healing or protection influences God's intervention?
I grew up rather fundamentalist Christian (Church of Christ) and have also studied religion and religious belief at various times throughout my academic career and I can tell you with certainty that...
...I have no idea.
On a serious note, though, there's quite a bit of variation in beliefs in modern Christianity about intercessory prayer. A person can't really say, "Christians believe X about prayer," because many denominations disagree about fundamental aspects of Christianity. If we took a Baptist, Southern Baptist, Methodist, Lutheran, Church of Christ, Disciple of Christ, etc. and put them all in a room together and asked them to explain what Christianity says about prayer, I doubt you'd get much agreement/consistency.
As others have noted, some Christians believe God grants prayer requests if it fits His plan, some Christians believe He will change His plan to grant requests, some believe that He gave us free will and there is only a "goal" but not a deterministic "plan" which means that He will grant prayers/miracles occasionally if the conditions are right, some are probably not really sure what they believe about it, etc.
A person can read an internet article(s) about the history/origins of prayer in Catholicism and Christianity, but that doesn't quite capture the variation and nuance that exists in real life (religion as it is actually preached and practiced) among all the Christian denominations throughout the world.
While I was growing up in the church, I was always kind of confused about intercessory prayer because I was always getting vague, somewhat conflicting messages about it from my preachers, Sunday School teachers, and high school Bible teacher (I graduated from a private Christian school). Back then, I didn't know the word for it ("intercessory") because my church didn't differentiate. Prayer is prayer, ya know?
There are various verses that Christian intercessory prayer beliefs are based on. One of the main ones is
Mark 11:24: "Therefore I tell you, whatever you ask in prayer, believe that you have received[a] it, and it will be yours."
There's also the "Ask and it shall be given unto you" verse from
Mark 7, which also implies that prayers will be answered. There are others, too, but those are good for this point.
However, there are also several verses which many Christians interpret as saying that the world operates "according to God's divine plan," like
Ephesians 1:11,
Jeremiah 29:11-14, and
2 Peter 3:9, just to name a few (there are also references to God having a plan in the OT, such as in Psalms).
There's also the issue of the differences in what is literally in the scriptures and what is actually taught from the pulpits. From my experience, it seems that a lot of Christians have an overall kind of inconsistent and fuzzy idea about prayer and God's plan, but they don't really give it much deep, critical examination (probably because doing so would unravel some logical conundrums).
I was taught that if we pray, God sometimes grants what we ask, but the details of how all that works was always glossed over, therefore always a bit fuzzy to me. I would be told "Everything that happens is according to God's plan" in one sermon, and then in the next week's sermon, I'd hear a lesson on "The Power of Prayer! Ask and it shall be given unto you." So, which is it? I don't know. It seemed that people I grew up with would apply whichever of those beliefs best fit the situation they were in and would allow them to feel more positive about it.
It never really made much logical sense to me (it still doesn't, which is one reason I'm not really religious anymore) and my confusion can be summed up by two flowchart pictures (
this one and
this one). Those aren't at all meant to belittle or dismiss anyone's beliefs about prayer, but they do highlight the logical inconsistencies in Christian prayer doctrine that I always struggled to reconcile.
Overall, my experiences while I was a Christian and my experiences with Christians from different denominations is that, even though there's a lot of variation and disagreement among Christians about prayer, and many of those beliefs are logically if not doctrinally contradictory, prayer brings them comfort. Some believe that their prayers really do lead to miracles, but I think a lot of them know that nothing will happen and they use the process of prayer as more of a tool to give comfort to themselves and others in times of need. Like a warm blanket in the cold.
If it helps them feel more connected to themselves, their loved ones, and their god, then I don't really see anything wrong with it and I'm glad they have a tool which helps them find peace.
But, for those who really do believe that prayer leads to healing miracles and they rely on prayer rather than medicine to heal their loved ones (kids), then I have a huge problem with it because
that belief leads to injury and death of innocent people.