Here's a brand new attempt at a funny joke about a goose:
Why did the goose go to the doctor?
Because it had a fowl cough!
Explanation: I set up the expectation that the goose is going to the doctor with a serious medical issue, like a human would. But the punchline subverts this expectation by using "fowl" (meaning related to or characteristic of geese) to reference the goose's species, but also to describe the sound of a cough, which is both a common medical symptom and a characteristic of a goose's honking noise. The humor comes from the unexpected twist on the wordplay, and the image of a goose, which is often depicted as a bit goofy or silly, having a cough that sounds like its own honk.