Theatre award, not the first received by a haggard columnist (5,4)
I believe the answer is:
agony aunt
'columnist' is the definition.
(I know that agony aunt is a type of newspaper columnist)
'theatre award not the first received by a haggard' is the wordplay.
'theatre award' becomes 'Tony' (as in the Tony Awards).
'not the first' means to remove the first letter.
'received by' indicates putting letters inside.
'haggard' becomes 'gaunt' (synonyms).
'tony' with its first letter taken off is 'ony'.
'a'+'gaunt'='agaunt'
'ony' put inside 'agaunt' is 'AGONY AUNT'.
(Other definitions for agony aunt that I've seen before include "Newspaper columnist who offers personal advice to readers" , "One like Madeley, e.g.," , "Columnist dealing with readers' letters" , "Newspaper problems columnist" , "sympathetic advisor" .)