I have a broken blood vessel in the ball of my hand. It is the size of a large pea and tender to the touch and aches constantly. Been there 3 weeks.

May need drainage. What describe sounds like something called a hematoma or trapped blood under the skin especially if the cause was secondary to trauma. Usually this would be raised with surrounding blue or yellow-green skin. If you have had for 3 weeks, this may require drainage or evacuation under local anesthesia. You should see a general or hand surgeon.