The most common and most likely answer to this crossword clue is the 5 plus 5 letter word BLOOD SPORT
I've seen this crossword clue in Independent
B
L
O
O
D
S
P
O
R
T
More clues leading to the result BLOOD SPORT
Buddy explains!
The straight part
We know that "deadly pursuit for some" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case deadly pursuit for some = blood sport.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.