Deputy cleaner cutting Frenchman's cost of upkeep (7,6)
I believe the answer is:
service charge
'cost of upkeep' is the definition.
'service charge' can be an answer for 'cost' (service charge is a kind of cost). I am not sure about the 'of upkeep' bit.
'deputy cleaner cutting frenchman's' is the wordplay.
I cannot really understand how this works, but
'deputy' could be 'vice' (as in vice-president) and 'vice' is found within the answer.
'cleaner' could be 'char' (charwoman is a type of cleaner) and 'char' is present in the answer.
'frenchman' could be 'serge' (Serge is a common French name) and 'serge' is located in the leftover letters.
No letters remain.
This explanation may well be incorrect...
Can you help me to learn more?
(Other definitions for service charge that I've seen before include "Restaurant may have one" , "Restaurant bill add-on" , "Addition to diner's bill" , "Additional restaurant fee" , "Part of bill" .)