The most common and most likely answer to this crossword clue is the 7 letter word SUBPLOT
I've seen this crossword clue in The Guardian
S
U
B
P
L
O
T
More clues leading to the result SUBPLOT
Buddy explains!

The straight part
We know that "a minor story" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case a minor story = subplot.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does SUBPLOT mean?
SUBPLOT noun- A plot within a story, subsidiary to the main plot.
- A subdivision of a plot of land, especially one used for an agricultural experiment.
- To provide (a story) with a subplot.