The most common and most likely answer to this crossword clue is the 3 plus 4 letter word BIG GAME
I've seen this crossword clue in The Times
B
I
G
G
A
M
E
Buddy explains!
The straight part
We know that "Pride, perhaps/a relegation six-pointer?" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case pride, perhaps/a relegation six-pointer? = big game.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.