Exercise, regularly painful hard work requisite for cook? (6,3)

I believe the answer is:

peanut oil

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'requisite for cook?' is the definition.
Although both the answer and definition are singular nouns, I cannot understand how one could define the other.

'exercise regularly painful hard work' is the wordplay.
'exercise' becomes 'pe' (Physical Education).
'regularly' indicates alternate letters (regularly take one letter, leave next etc.).
'hard work' becomes 'toil' (I've seen this before).
The alternating letters of 'painful' are 'anu'.
'pe'+'anu'+'toil'='PEANUT OIL'

Can you help me to learn more?

(Another definition for peanut oil that I've seen is " substance inside soap?".)

I've seen this clue in The Times.

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