Suggested answer:
(1) The following nouns/NPs follow immediately after the preposition at in the corpus: a distance, the market, a guess, and war. Examples from the corpus:
For those who followed fashion at a distance, ...
...meet the Frenchman at the market.
... twenty at a guess.
Well, I am at war with no one.

(2) The following nouns/NPs follow immediately after the preposition on in the corpus: holiday, the market, a diet, impulse, and war. Examples from the corpus:
Again we were on holiday...
...these Jacobean properties don't come on the market every day.
Well, I've been on a diet since I was nineteen...
Mostly I would just see them by accident and go into them on impulse.
If Umuofia decided on war, all would be well.

(3) The following nouns/NPs combine both with at and on: the market and war. In the case of the market, at introduces a PP of location, the market has the meaning of "a place where people meet for the purpose of buying and selling merchandise". If preceded by on, the market typically has the meaning of "available for purchase" or "trade"/"business".

When preceded by at, war is part of a PP, and the meaning is "in the middle of an (armed) conflict". In cases where on precedes war, on is typically part of the VP (i.e. part of a phrasal verb), as in If Umuofia decided on war, all would be well.
War has its core meaning of "(armed) conflict" between two or more parties.

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Read about prepositions here
Terminology: Noun Phrase (NP), preposition, Prepositional Phrase (PP), phrasal verb