County charity's first to go broke (4-3-3)
I believe the answer is:
down-and-out
'broke' is the definition.
(I've seen this in another clue)
'county charity's first to go' is the wordplay.
'county' becomes 'down' (County Down in Northern Ireland).
'charity' becomes 'handout' (handout is a kind of charity).
'first to go' means to remove the first letter.
'handout' with its first letter taken off is 'andout'.
'down'+'andout'='DOWN-AND-OUT'
(Other definitions for down-and-out that I've seen before include "Impoverished (and unconscious?)" , "Impoverished - destitute person" .)