The most common and most likely answer to this crossword clue is the 4 letter word WEEP
I've seen this crossword clue in The Times
W
E
E
P
Buddy explains!

The straight part
We know that "Keen" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case keen = weep.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.
What does WEEP mean?
WEEP noun- A session of crying.
- To cry; shed tears.
- To lament; to complain.
- (of a wound or sore) To produce secretions.
- To flow in drops; to run in drops.
- To hang the branches, as if in sorrow; to be pendent; to droop; said of a plant or its branches.
- To weep over; to bewail.
- A lapwing; wipe, especially, a northern lapwing (Vanellus vanellus).