The T-shirt is sold in a store, for some amount, say 12$$12. Of the 20$$20 you've given your friend, 12$$12 will go to the store, and 8$$8 will go to your friend. You can say:
- Your friend has received a 8$an $8 commission to buy a 12$$12 T-shirt for you;
- You gave your friend 20$$20 to buy you a T-shirt, and told them to keep the change.
The second one is probably more accurate, because you don't need to know how much the T-shirt was sold for; in fact, your friend might have had to make a choice, for instance between a 10$$10 T-shirt at a far-away shop or a 14$$14 T-shirt at a close shop. The choice is theirs and you don't care either way, since you always end up paying a total of 20$$20.
If you say "I paid my friend 20$$20 to buy a T-shirt", it's pretty confusing, because it sounds like those 20$$20 will all go to your friend, and do not include the price of the T-shirt.