You know that an お願い is a request, and that 事 is a "thing" or "action." So, this is a "request action," in a sense. A better way to say that is one's wish or a prayer, because that's what you're doing when you do the action of requesting something.
The readings come from the words お願い and 事. If you know how to read those you should be able to read this word too. Just remember that 事 is rendaku'd, so it's read as ごと.
I think your prayers will be answered soon.
When Koichi goes to sleep he always keeps a piece of paper with his wishes written on it nearby.
Japanese people visit shrines on important occasions not only to make wishes but also to offer thanks.