Heilbroner. This Is the House Where Jack Lives

This Is the House Where Jack Lives is a children's book by Joan Heilbroner, with art by Aliki, published in 1962. It is available at the Internet Archive.

Here is a YouTube read-along:

This is a nice example of a super-simple "Jack" story; like the traditional "Jack" it starts off with the short lines, then adds rhyme, and it's told cumulatively. And wouldn't you know it: a cat's the troublemaker!

This is the cat that liked to play,
That jumped on the girl so fat and gay,
That shook the mop so old and gray,
That bumped the man,
That held the pail,
That fell on the boat,
That walked the dog,
That lives in the house
Where Jack lives.

Other missteps and mishaps occur because of that cat... and in the end we finally meet Jack and find out he is a troublemaker too: he lets the bath water keep running too long!

Not the best "Jack" story, but it's a good example of the formula, something that might give you some ideas of your own.