The most common and most likely answer to this crossword clue is the 4 letter word MAWS
I've seen this crossword clue in Financial Times
M
A
W
S
Buddy explains!
The straight part
We know that "greedy mouths" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case greedy mouths = maws.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 MAWS mean?
MAWS noun- The stomach, especially of an animal.
- The upper digestive tract (where food enters the body), especially the mouth and jaws of a fearsome and ravenous creature.
- Any large, insatiable or perilous opening.
- Appetite; inclination.
- Mother.
- A gull.