The most common and most likely answer to this crossword clue is the 7 letter word BLUBBER
I've seen this crossword clue in The Daily Telegraph
B
L
U
B
B
E
R
Buddy explains!
The straight part
We know that "Fat/crybaby" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case fat/crybaby = blubber.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 BLUBBER mean?
BLUBBER noun- A fatty layer of adipose tissue found immediately beneath the epidermis.
- Fatty tissue.
- The thick coat of fat worn by many Arctic animals, such as sea lions, and Antarctic animals, such as penguins; used to insulate warmth in the animal's body.
- A bubble.
- To make noises or broken words while crying.
- To swell or disfigure (the face) with weeping; to wet with tears.