A dear friend once gave a personal testimony about the power of praying specifically, that is, of asking the Lord to attend to particular needs. It was her experience that knowing enough about a situation to ask the Lord very clearly for a particular need seemed to help her prayers be more "effective" in some way.
There is no way we can "bribe" or "control" God. We always pray "thy will be done." However, because God is love, it seems (subjectively speaking) that the following is true: The self-giving love we have for someone can reflect God's love and God's grace in prayer more effectively than, for instance, when we pray merely dutifully "for the sick" (or for a particular person with whom there is only a superficial relationship).
I think prayer is closely connected with relationships. God is the master of relationships. In fact, God's own being includes relationship. When, therefore, we allow ourselves to be affected by the needs or trials of others, and when we are doing our best to meet those needs, and when we're not keeping those needs all to ourselves, healing seems to flow more freely.
If someone reading this could check or challenge this in biblical terms, I'd appreciate it. I see this principle in the life of Christ, as he responded to needs that were brought to him or that he saw. I know of no time when Christ somehow spiritually discerned that a person was in need and went to seek them out alone when they were in hiding. Even the woman at the well was out in the open when Jesus came to her. The desperate need for caring and love was met when it wasn't hidden away.
So, I'd like to encourage you to bring your burdens to others, to allow others to care for you, and then to go together to the Lord in prayer. If you know someone in need, go to them and let them know you care. That caring is, I think, a vital step in praying for the Lord's intervention. Don't keep it too quiet, too private. No need to broadcast all over, but at least bring in those you know who care.
For a bit more on this, see Pray Y'all.