The most common and most likely answer to this crossword clue is the 8 letter word OPTIMISE
I've seen this crossword clue in The Daily Telegraph
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More clues leading to the result OPTIMISE
Buddy explains!

The straight part
We know that "Make best use of" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case make best use of = optimise.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.
Crossword Answer definitions
OPTIMISE verb- (originally intransitive) To act optimistically or as an optimist.
- To make (something) optimal.
- To make (something) more efficient, such as a computer program.
- To become optimal.