The most common and most likely answer to this crossword clue is the 4 letter word ITCH
I've seen this crossword clue in The Times
I
T
C
H
More clues leading to the result ITCH
- Pain when running, having to lose stone? That’s an irritation
- Long hold-up after the start
- Longing to thumb a lift in part of London?
- East End couple: their problem after seven years?
- Female dog losing head in desire
- Long pain after exercising to lose stone
- Crone a non-starter in desire
- Long delay, but heading off
- Long throw has opener dismissed
- Chit-chat conceals irritation
Buddy explains!

The straight part
We know that "Burning" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case burning = itch.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 ITCH mean?
ITCH noun- A sensation felt on an area of the skin that causes a person or animal to want to scratch.
- A constant teasing desire or want.
- To feel itchy; to feel a need to be scratched.
- To have a constant, teasing urge; to feel strongly motivated; to want or desire something.
- To cause to feel an itch.
- To scratch or rub so as to relieve an itch.