The most common and most likely answer to this crossword clue is the 9 letter word OVERPLAYS
I've seen this crossword clue in Financial Times
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Buddy explains!

The straight part
We know that "Exaggerates importance of" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case exaggerates importance of = overplays.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 OVERPLAYS mean?
OVERPLAYS verb- To overdo or overact one's effect or role.
- To play (a song or record) too frequently.
- To overestimate one's strength in a game or event, which ultimately may end in a defeat.
- To accidentally hit (one's golf ball) beyond "the green".