When we pray for something and God responds, I refer to that as an intercession or as intercessional prayer.
Some intercessions are obvious. the classic prayer for the presence of the Spirit, seeking nothing more, and obtaining the sweet calmness and light of the Spirit, is a prayer seeking intercession and obtaining it. But it is easy for that prayer to be a surrender to lowered expectations rather than a beginning.
The next type of intercessory prayer is for knowledge of God's will and the strength and courage to embrace and carry it out. For my wife, that meant going to graduate school. The CRNA graduate program was mind numbingly difficult and a miracle, given our circumstances. (Yes, I admire and respect my wife). But it was also made possible by the intercession of God's answering her prayer and giving her knowledge, strength and courage.
But, the intercession most of us want is for God to intervene and change things -- changing our lives or the lives of others. Job seeking, blessings seeking healing, changes of heart for ourselves or others, deliverance from peril, those are the sorts of intercession we seek. It is not so much that there are no atheists in fox holes as there are few who feel like a foxhole is the place to surrender to destiny.
God responds in all of these requests for intercession. While sometimes we may "Thank God for Unanswered Prayers" (to quote a song about the blessing of God saying no when getting what we wanted would have been a disaster), some times God says "yes." I've had both happen -- God say no, and God say yes, both when I expected the other answer.
I write about seeking to have God say yes next, which reprises back to the courage to embrace God's will as well as the faith and strength to seek to affect it, which touches upon one of the mysteries -- why is life painful (among other mysteries).