Hasten the process of former gym taking on correction (8)

I believe the answer is:

expedite

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'hasten the process' is the definition.
'expedite' can be an answer for 'hasten' (synonyms). I'm unsure of the 'the process' bit.

'former gym taking on correction' is the wordplay.
'former' becomes 'ex' (prefix meaning former or onetime).
'gym' becomes 'pe' (short for Physical Education).
'taking on' indicates putting letters inside.
'correction' becomes 'edit' (I've seen this in another clue).
'pe' going around 'edit' is 'pedite'.
'ex'+'pedite'='EXPEDITE'

'of' acts as a link.

Can you help me to learn more?

(Other definitions for expedite that I've seen before include "Hasten (something)" , "Rush" , "Assist the progress of" , "Hasten (process)" , "Speed up progress (of)" .)

I've seen this clue in The Irish Times.

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