The most common and most likely answer to this crossword clue is the 8 plus 3 letter word PRODIGAL SON
I've seen this crossword clue in The Times
P
R
O
D
I
G
A
L
S
O
N
Buddy explains!
The straight part
We know that "Extravagant relation" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case extravagant relation = prodigal son.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.