Dear Mr. Goldberg,
Dante, in Purgatorio, and, I assume, other places mentions the longstanding Catholic belief that temporal punishment for sin is a specific, albeit unknown, quantity and that prayer, fasting, and sacrifice for others can lessen that temporal punishment (for the others as well as oneself since such prayer is an act of charity). While it is hard to do better than God himself suffering for the remission of our sins, it is nonetheless generally accepted that such prayer can be efficacious and since there is a specific quantity of time at issue, it makes sense that more people praying would be more helpful. Dante is certainly not the only source for this proposition, but he is the one with which I am most familiar. I’m pretty sure that you will find something similar in Thomas.
But I see very little logic behind a lot of people praying for, for example, good weather. When I hear such things in Catholic circles, I attribute it to the ever Protestant-toward drift of the Church, not to anything that is a substantial part of the tradition or that stands on its own two feet.