None, right? Not so, in a manner of speaking.
A friend of mine recently asked about the difference between hope and prayer and what these intentions would mean with a rejection of the supernatural. ‘Prayer’, in modern vernacular, has developed two meanings: prayer as meditation (presumably the original meaning, e.g. “the monks prayed for hours”) and prayer as hope or intent (the common meaning, e.g. “I pray you’ll have a speedy recovery”).
Prayer as meditation is particularly fascinating when compared with other popular meditative philosophies like Buddhism, which lack a supernatural agent. The broad idea in many forms of meditation is not to alter the physical world but to accept new perspectives through your own agency which would effectively alter your reality, e.g. death is both something one fears as well as a natural system playing out a normal course — meaning is in the eye of the beholder.
So when we ask what effect prayer has on reality, the obvious answer to our physical, empirical reality is absolutely none, save for the calming effects confirmed by neuroscience and how that might affect your communal actions throughout the day. But we must give credit to prayer’s ability to alter one’s internal perspective, at least as much as we credit music with affecting our moods.
But if hope and prayer in a Godless world merely shift our own perspective, what effect would praying for someone actually have on them? None, right? The issue at hand is your lack of agency over their thoughts: you cannot force another individual to meditate and realize alternative perspectives to their otherwise downed reality.
Strangely enough, this is the effect one does have when they convince someone that God does exist, that God has a plan for them, etc., albeit in a disingenuous fashion.