The most common and most likely answer to this crossword clue is the 5 plus 5 letter word CHUCK STEAK
I've seen this crossword clue in The Daily Telegraph
C
H
U
C
K
S
T
E
A
K
Buddy explains!
The straight part
We know that "not the finest cut of meat" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case not the finest cut of meat = chuck steak.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.