The most common and most likely answer to this crossword clue is the 13 letter word ONE-UPMANSHIP
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More clues leading to the result ONE-UPMANSHIP
Buddy explains!

The straight part
We know that "Maintaining advantage" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case maintaining advantage = one-upmanship.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 ONE-UPMANSHIP mean?
ONE-UPMANSHIP noun- The art or practice of successively outdoing a competitor.
- A succession of instances of outdoing a competitor.