Paper starting with cryptic has part of mine (5)
I believe the answer is:
shaft
'part of mine' is the definition.
('shaft' is part of 'mine')
'paper starting with cryptic has' is the wordplay.
'paper' becomes 'FT' (newspaper the FT, Financial Times).
'starting with' means one lot of letters go next to another.
'cryptic' indicates an anagram (a cryptic or obscure order of the letters).
'has' is an anagram of 'sha'.
'ft' put after 'sha' is 'SHAFT'.
(Other definitions for shaft that I've seen before include "Vertical passageway, into a mine say" , "An arrow or a mine passage" , "A little light" , "Body of an arrow" , "Ray (of sunlight)" .)