People who must work with clients in high stress situations learn to "compartmentalize." Basically, you think about work while at work, and home while at home. Honestly, I would "forget" my customers the minute they walked out the door. I might have notes on things that I need to follow up on for them, but they are out of my mind. And they needed to be, because the next customer was walking in, and I needed to be 100% focused on them.
As one individual you are not going to be able to save the world. So, for example: as tempting as it might be to give them the money they need to go buy groceries, you accomplish a lot more if you let them know about the free food pantries, or, better yet, teach them how to find the free food pantries on their own.
I recommend you research compartmentalizing. It's an essential tool, and will help you be a successful psychologist!