The most common and most likely answer to this crossword clue is the 6 letter word REGRET
I've seen this crossword clue in Independent
R
E
G
R
E
T
More clues leading to the result REGRET
- Fret about wading bird seen by river
- Rue de Pierre Gretzky
- Feel remorse having run over bird
- Deplore being endlessly cruel
- Sadness as chap brought back tailless bird
- Be sorry some winter germs turned up
- Feel sorry to be winger right on the left
- Feel sorry about Garbo, briefly
- Bemoan what’s in the heart of bitter German from the East
- River bird makes one feel sorry
Buddy explains!

The straight part
We know that "Feel remorse" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case feel remorse = regret.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 REGRET mean?
REGRET noun- Emotional pain on account of something done or experienced in the past, with a wish that it had been different; a looking back with dissatisfaction or with longing.
- Dislike; aversion.
- To feel sorry about (a thing that has or has not happened), afterthink: to wish that a thing had not happened, that something else had happened instead.
- (more generally) To feel sorry about (any thing).
- To miss; to feel the loss or absence of.