The most common and most likely answer to this crossword clue is the 4 plus 2 plus 1 plus 4 letter word FREE AS A BIRD
I've seen this crossword clue in The Times
F
R
E
E
A
S
A
B
I
R
D
Buddy explains!
The straight part
We know that "not kept in check" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case not kept in check = free as a bird.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.