Jesse Pinkman first meets Andrea at a Narcotics Anonymous meeting in which his original intention was to sell her some methamphetamine. However, after finding she is a mother of a six-year old son named Brock, he decides against it. Instead, he starts a relationship with Andrea and becomes friends with Brock ("Abiquiú").
Thanks to financial support from Jesse, Andrea and Brock can afford to move to a better house and leave the violent neighborhood in which they lived ("Thirty-Eight Snub"). Jesse was away from them, but still frequently sent money through Saul Goodman so that they could have a good life ("Hermanos").
Some time later, Jesse reunites with Andrea and plays video games with her and Brock ("Crawl Space"). Soon after, Brock is rushed to the hospital after he suddenly becomes ill by some unknown cause. Jesse believes he's been poisoned with the ricin that he had concealed in his pack of cigarettes, which was originally intended for Gustavo Fring. Jesse seeks Walter White thinking he was responsible for ricin poisoning, but Walt leads Jesse to think that Gus poisoned Brock, thus giving Jesse a motive to be against Gus ("End Times").
After Walt causes Gus's demise and he and Jesse set the superlab up in flames, Walter takes Jesse back to the hospital to learn the status of Brock. Jesse learns that the doctors found that Brock was not poisoned with ricin, but with poisonous berries of a plant called Lily of the Valley. It is strongly implied that Walt was responsible for poisoning Brock with the poisonous berries, as he wished to manipulate Jesse and turn him against Gus. A potted Lily of the Valley plant can be seen sitting by Walter's back pool ("Face Off").
Brock leaves the hospital and is completely recovered. Jesse, Andrea and Brock later have dinner at Jesse's house with Walter. Interestingly, Walter and Brock sit on the couch for a moment and look briefly: he doesn't seem to recognize Walt as the man who poisoned him, and is instead more interested in his Playstation ("Hazard Pay"). Walt is clearly not comfortable being with him.
Jesse was initially worried about the missing ricin cigarette being out in the open for someone to find and get sick from (like "a child," Jesse suggests), so Walt creates a fake ricin cigarette filled with salt instead of ricin and plants it in Jesse's house for him to find later, relieving his concerns ("Madrigal").
Walter, probably thinking of the risk caused by Jesse's contact with Andrea and Brock, manipulates him again by telling him that their relationship is not going anywhere if he has to keep lying about his business (drug trafficking). With this in mind, Jesse starts to feel bad being with Andrea, and decides to end the relationship and stop seeing her and Brock ("Hazard Pay").
Jesse later finds out that Saul had Huell steal the ricin cigarette from Jesse's pocket, and that Walt was responsible for Brock's poisoning. ("Confessions")
Brock's ultimate fate is unknown after the murder of his mother, Andrea, but it could be presumed that his grandmother took him in.