Dead tree damaged new leader’s address (6,3)
I believe the answer is:
number ten
'address' is the definition.
(home of British Prime Minister)
'dead tree damaged new leader's' is the wordplay.
'dead' becomes 'numb' (as in a dead arm).
'damaged' indicates anagramming the letters.
'leader's' suggests taking the first letters.
The first letter of 'new' is 'n'.
'tree' is an anagram of 'erte'.
'numb'+'erte'+'n'='NUMBER TEN'
(Other definitions for number ten that I've seen before include "where the PM can be found?" , "political residence" , "Prime Minister's official residence" , "Stand-off" , "Figure on the British PM's door" .)