As an insignificant person, hard to become PM (7)
I believe the answer is:
asquith
'pm' is the definition.
(H. H. Asquith, UK prime minister 1908-1916)
'as an insignificant person hard' is the wordplay.
'an insignificant person' becomes 'quit' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'hard' becomes 'h' (abbreviation used in pencil classifications).
'as'+'quit'+'h'='ASQUITH'
'to become' acts as a link.
(Other definitions for asquith that I've seen before include "old cabinet maker" , "Herbert - - , WW1 PM" , "Old PM" , "PM in 1914" , "Politician" .)