Can, perhaps, with old sauce (4)

I believe the answer is:

mayo

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'sauce' is the definition.
(I know that mayo is a type of sauce)

'can perhaps with old' is the wordplay.
'can perhaps' becomes 'may' (synonyms. I am not sure about the 'perhaps' bit.).
'with' is a charade indicator (letters next to each other).
'old' becomes 'o' (abbreviation as in OT - Old Testament).
'may'+'o'='MAYO'

Can you help me to learn more?

(Other definitions for mayo that I've seen before include "radio presenter" , "County of north-west Ireland" , "Eg, Braeburn or Cox" , "Irish county - sauce (abbr)" , "Irish county -- salad cream" .)

I've seen this clue in The Telegraph.

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