The most common and most likely answer to this crossword clue is the 9 plus 5 letter word CROCODILE TEARS
I've seen this crossword clue in Financial Times
C
R
O
C
O
D
I
L
E
T
E
A
R
S
More clues leading to the result CROCODILE TEARS
- Children paired for walking races feigned distress
- Playing role, cad cries to fake grief
- Put on grief crier acted – so low, almost perverse
- False emotion from which actors recoiled in play
- Lousy actors recoiled in false show of emotion
- Exuberant actors recoiled in false show of emotion
- Line of pupils rushes in simulated distress
- Hypocritical grief from snappy types trying to appear wet?
Buddy explains!
The straight part
We know that "Put on grief" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case put on grief = crocodile tears.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.