The most common and most likely answer to this crossword clue is the 6 letter word BEMOAN
I've seen this crossword clue in The Times
B
E
M
O
A
N
More clues leading to the result BEMOAN
- Lament of man about extinct bird
- Deplore embargo covering moody music
- Grieve for bird that disappeared inside mountain
- Lament extinct bird in Scottish mountain
- Grumble about head, getting hold of doctor
- Lament hit on head, calling in doctor
- Complain about Bill’s partner embracing old bird
- Regret doctor taking pulse? On the contrary
- Complain about Menace initially in Beano comic
- Live with man without love and complain
Buddy explains!

The straight part
We know that "Grieve" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case grieve = bemoan.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 BEMOAN mean?
BEMOAN verb- To moan or complain about (something).
- To be dismayed or worried about (someone), particularly because of their situation or what has happened to them.