Cleaner to rush outside for additional cost (9)
I believe the answer is:
surcharge
'additional cost' is the definition.
(I've seen this before)
'cleaner to rush outside' is the wordplay.
'cleaner' becomes 'char' (charwoman is a type of cleaner).
'to rush' becomes 'surge' (synonyms).
'outside' indicates putting letters inside (some letters go outside others).
'char' inserted within 'surge' is 'SURCHARGE'.
'for' acts as a link.
(Other definitions for surcharge that I've seen before include "extra amount" , "Additional item on bill" , "Additional sum demanded" , "Additional payment" , "Additional cost" .)