The most common and most likely answer to this crossword clue is the 5 letter word SINGE
I've seen this crossword clue in The Guardian
S
I
N
G
E
More clues leading to the result SINGE
- Slightly burn grouse primarily served during function
- Performer getting cut, a superficial mark
- Unmarried, not having pound to burn
- Opera performer finally fails, becoming char
- Evidence of some Burns found in poems in general
- Vocalist briefly exhibiting a touch of fire
- Char’s function to keep cleaning, ultimately
- Char in French capuchin?
- Moers in Germany accommodates char
- Pipe needs fire, not wood, to burn
Buddy explains!

The straight part
We know that "burn" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case burn = singe.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 SINGE mean?
SINGE noun- A burning of the surface; a slight burn.
- To burn slightly.
- To remove the nap of (cloth), by passing it rapidly over a red-hot bar, or over a flame, preliminary to dyeing it.
- To remove the hair or down from (a plucked chicken, etc.) by passing it over a flame.