The most common and most likely answer to this crossword clue is the 4 letter word HONE
H
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More clues leading to the result HONE
Buddy explains!

The straight part
We know that "Get increasingly keen" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case get increasingly keen = hone.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 HONE mean?
HONE noun- A sharpening stone composed of extra-fine grit used for removing the burr or curl from the blade of a razor or some other edge tool.
- A machine tool used in the manufacture of precision bores.
- To sharpen with a hone; to whet.
- To use a hone to produce a precision bore.
- To refine or master (a skill).
- To make more acute, intense, or effective.
- A kind of swelling in the cheek.
- To grumble.
- To pine, lament, or long.