Press chief possibly cycled up carrying it (6)
I believe the answer is:
editor
'press chief' is the definition.
(chief of a newspaper)
'cycled up carrying it' is the wordplay.
'cycled' becomes 'rode' (I've seen this before).
'up' says the letters should be written backwards (in down clue: letters go upwards).
'carrying' is an insertion indicator.
'rode' written backwards gives 'edor'.
'edor' placed around 'it' is 'EDITOR'.
'possibly' is the link.
(Other definitions for editor that I've seen before include "Reviser" , "Rag worker" , "Press fellow" , "Wapping boss" , "Person responsible for publication;" .)