As far as I can remember, I've only really attempted three kinds of petitionary prayer on my own since childhood:
1. The "please help me find my keys" prayer that mostly just calms me down enough to remember where they were
2. The "I am desperate to fall asleep after hours of insomnia so I'm going to lie here with my eyes closed and ask God to bless literally everyone I can think of" prayer where I try to list every person I've ever known until sleep comes.
Neither of those remotely qualify as turning to God without turning away from self and unless there is some dark consequence to me finding my keys, they don't seem to apply to the question I asked last night about one's responsibilities when asking God for something*.
3. Sometimes, when I'm passing a scene of an accident or something, I find myself thinking "Please let that person be OK" but that's almost a reflex, not a thought out and considered sort of prayer, so I don't know that it counts. Yet for all I know, a serial killer is in that ambulance, so I can't say that it's entirely divorced from these issues.
I've also been known to pray a "Hey, if there's anyone out here who can hear this, thank you so much for what just happened, I am so very grateful for all the wonderful gifts in my life," though I don't do that one often enough. I sometimes attempt more contemplative sorts of prayers, though I worry that my short attention span will be a problem. If there is a God of the prayer-listening variety, I hope God is understanding about prayers than begin meditative and wander off into earthly concerns and just plain thinking over things.
*The concept of "blessing" is, I think, sufficiently diffuse and open to God's interpretation that I don't think any harm can come of it, should it be answered.