Punky's dream is to become an astronaut when she grows up. When she sees the explosion of the Challenger Space Shuttle on TV, she is devastated.

After a tearful reunion between Punky and Henry, the Buckworths plan to move to Europe. Henry, however, is working as hard as possible to rebuild his business and win Punky back.

Simon Chillings performs a Rorschach inkblot test on the fake Punky, and based upon the results, he places the real Punky with a wealthy family. However, she does not take to high society very well.

Punky's friends visit her at Fenster Hall. Hoping that Simon Chillings won't notice (and by taking his glasses), Margaux impersonates Punky so that the real Punky can visit Henry in the hospital.

Just as Henry was planning to adopt Punky, a Fenster Hall bureaucrat named Simon P. Chillings rescinds his custody citing his age, his illness, and his lack of a steady income. Mike Fulton tries to change his mind but to no avail.

Henry's studio burns down; with no insurance and a mountain of bills he can't pay, he suffers a bleeding ulcer and is hospitalized.

Henry reluctantly buys a new refrigerator and puts the old one in the backyard. Mike teaches CPR to the children in class. Allen goofs off and is sent to the principal's office. When the children are playing Hide and Seek, Cherie hides in the fridge and suffocates. Allen doesn't know what to do, so it's up to Punky and Margaux to save their friend's life.

Punky receives a remote-controlled race car as a gift. She wants to take it to the racing track, but they do not allow girls to compete there.

Punky is afraid that Henry will be the next victim of a widely publicized murder spree.

During the Christmas shopping season, Punky sees a friend shoplifting merchandise and decides to steal a present she can't afford for Henry. She regrets this when she accidentally switches bags with Mike, who ends up in prison.

Punky and Cherie learn that a girl who moved into the building recently was kidnapped by her divorced father, and they are both living in the building under assumed names.

Punky befriends the school janitor who, in spite of her learning disability, is a gifted violin player.

On Halloween, Punky raises the ire of Mrs. Peevey, a cantankerous old woman who lives in the building, simply by trying to be her friend.

Punky has something in common with her teacher, Mike Fulton: they were both abandoned by their mothers. They work together to try and find the whereabouts of their respective families.

Punky and Cherie are admitted into a clique, only to find that they use drugs and want them to do so.

The children do battle with an evil spirit, when the ghost of an Indian princess appears to tell Punky that it must be fought with love.

Punky and the gang go on a camping trip. When the children get lost in a cave, Punky tells a ghost story about an evil spirit who threatens the lake.

The children enroll in a tap dancing class taught by Jersey Janet, a former Broadway dancer. Punky is not very good, and she quits the class when her skills do not improve.

With Mrs. Johnson's assistance, Punky and Cherie go into the babysitting business. A woman leaves her baby with them, but when the three of them take it out for a stroll, they return home with another infant.

Punky and Allen cheat on their geography test.

Punky wants to build a tree house behind her apartment building, but she has trouble getting people to help her.

When Punky bears the brunt of the school bully's aggression, she asks professional boxer Marvin Hagler for advice on self-defense.