I managed to find the complete exercise on the Internet, given that you haven't provided us with all options given to you.
1. cogitating = contemplating
To cogitate means to think about something (cogito in Latin literally means 'to think). Contemplating means to mull over a couple (or a lot) of options which is what Mr. Gamefield is doing in the excerpt above - he is thinking of ways to pay his rent.
2. cudgeling = beating
A cudgel is a short stick which you can use as a weapon. So if you cudgel something or someone, it means that you are beating them with a cudgel, because it is a thick stick, almost like a bat. So this person was metaphorically cudgeling his brain, but literally cudgeling his donkey.
3. regaled = rewarded
The verb to regale has two meanings - it can either mean 'to entertain someone,' or it can mean 'to supply someone with something.' Here, the latter meaning is used - the donkey thought he would be given a cabbage-stalk or two for his hard work.
The story is thus a mystery.
Explanation:
A mystery has some of the following characteristics-
It is usually solved by the end of the tale or the story
-It gives clues throughout the plot that keep on increasing in terms of how much they reveal.
-The mystery is usually an answer to a problem that has been plaguing the person who is asking for a long time.
Without these characteristics a story is usually not a mystery and they can be said to be mostly universal thus in terms of being mystery.
These are the ones that primarily define the genre.
