Speed up progress of divorcee with exercise and new diet (8)
I believe the answer is:
expedite
'speed up progress' is the definition.
(I've seen this before)
'divorcee with exercise and new diet' is the wordplay.
'divorcee' becomes 'ex' (I've seen this in other clues).
'with' means one lot of letters go next to another.
'exercise' becomes 'pe' (Physical Education).
'and' says to put letters next to each other.
'new' indicates an anagram.
'diet' with letters rearranged gives 'dite'.
'ex'+'pe'+'dite'='EXPEDITE'
'of' acts as a link.
(Other definitions for expedite that I've seen before include "Hasten (process)" , "Hasten, facilitate" , "Assist the progress of" , "Hurry" , "Speed up progress (of)" .)