You have to collect briefcases to extend the time and earn money. Earn more money then the C-rank before time runs out. You unlock more Hot Potato side missions and Weapon Caches for A-rank or better. There are 15 total.
TIP: I found the Super Bike the best vehicle for Hot Potato because it has good acceleration and its good for driving through tight spaces like alleys. Try to remember where all the packages are because some of them are hard to find and it makes it a lot easier if you know where they all are.