Icheb passes "Early Starfleet history". Then Q and his kid turn up and Janeway gets to look horrified.
Kid Q is on vacation. Before Janeway can give him rules, he sods off. Later, he turns up on the bridge, and says he's bored and goes to turn Engineering into a rave. Did Janeway just swear?
Tuvok suggests ignoring him in the hopes he'll go away. So the first thing he does is make Seven of Nine's clothes disappear and gives the computer a personality.
Neelix suggests that he parent kid Q a bit. This results in Kid Q making his vocal bits disappear, something I'm sure Tuvok has wanted to do for a while now. Then he magics up three Borg cubes to hassle Voyager. Fortunately Q turns up in time to tell him off for provoking the Borg.
Janeway complains to Q, and he complains that Kid Q has been up to all sorts of mischief. Janeway suggests parenting.
Q's supposed to be omnipotent? Why's he aging?
Q pops up in Janeway's bath, and they have a frank talk about parenting in the bubble bath. He turns Kid Q into an ameoba of some sort for a while as a punishment and takes away his powers and leaves after giving him an ultimatum - shape up in 7 days or become an amoeba permanently
Janeway gives Kid Q some quarters and sets the crew educating him. Chakotay sets him to a diplomatic mission getting a bunch of species to agree over mining rights, but he cheats, of course. He also gets Icheb to write an essay for him. Janeway grumps at him for cheating, and forces him to agree to play it straight. And surprisingly he does, and somehow changes his personality overnight, and Janeway lets him have a break taking a flying lesson in the shuttle with Paris and Icheb.
Q turns up for an update, and Kid Q gives a reading of his essay. Q isn't impressed, and Janeway is peeved at him.
Janeway gives Kid Q a peptalk. Later on he and Icheb correct an ion imbalance in the Flyer, and then Kid Q steals the Flyer and shoots off into a spacial rift to another system. Then a vessel turns up and tries to take them into custody for violating their space. Kid Q opens another rift, but Icheb is zapped, and Kid Q returns to Voyager.
HoloDoc can't treat him without knowing more about the weapon used. Janeway says they need to go back, apologise, and get the info they need. Janeway and Kid Q return to the other system, but the alien says Janeway will get punished for his actions. Kid Q pleads that he be punished instead.... and the alien turns out to be Q making another test.
The Q continuum passes judgement... and sentences Kid Q to remain human.
Kid Q asks Janeway if he can remain on Voyager. Then Q turns up again, and says he appealed and won. After Kid Q leaves, he admits to Janeway that he had to promise to stay with the kid, and hands her a shortcut which will trim a few years off the journey home. Teh end.
I think if you look back over season 7, you can just about see the point where the writers stopped trying.