The most common and most likely answer to this crossword clue is the 6 letter word FESTER
I've seen this crossword clue in The Times
F
E
S
T
E
R
More clues leading to the result FESTER
- Fine organic compound's gradually become bitter
- Banqueter ignoring a brew
- Some of life’s terrors get worse
- Clown and king go off
- Shakespearean fool has little right to be sore
- One enjoying a good meal after removing a small ulcer
- Inflamed sore in the giraffe's terrible
- Way to break free, free of rot
- Way to stop payment by year end — it will rankle
- Way to break free perhaps and rot
Buddy explains!

The straight part
We know that "gradually become bitter" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case gradually become bitter = fester.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 FESTER mean?
FESTER noun- A fistula.
- A sore or an ulcer of the skin.
- The condition of something that festers; a festering; a festerment.
- To become septic; to become rotten.
- To worsen, especially due to lack of attention.
- To cause to fester or rankle.