The most common and most likely answer to this crossword clue is the 5 letter word OUTDO
O
U
T
D
O
Buddy explains!
The straight part
We know that "Run circles round/barbecue" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case run circles round/barbecue = outdo.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 OUTDO mean?
OUTDO verb- To excel; go beyond in performance; surpass.