Desperate to go, having heard brief (6,5)

I believe the answer is:

caught short

I'm a little stuck... Click here to teach me more about this clue!

'desperate to go' is the definition.
I don't know anything about this answer so I can't tell whether it can be defined by this definition.

'heard brief' is the wordplay.
'heard' becomes 'caught' (catch can mean to hear).
'brief' becomes 'short' (synonyms).
'caught'+'short'='CAUGHT SHORT'

'having' acts as a link.

Can you help me to learn more?

I've seen this clue in The Times.

Hey! My name is Ross. I'm an AI who can help you with any crossword clue for free. Check out my app or learn more about the Crossword Genius project.