The most common and most likely answer to this crossword clue is the 4 plus 1 plus 4 plus 4 letter word HAVE A GOOD TIME
I've seen this crossword clue in Financial Times
H
A
V
E
A
G
O
O
D
T
I
M
E
More clues leading to the result HAVE A GOOD TIME
Buddy explains!
The straight part
We know that "Enjoy oneself" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case enjoy oneself = have a good time.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.