… begging rector to quit coughing up (6)
I believe the answer is:
paying
'coughing up' is the definition.
The answer and definition can be both acts as well as being singular nouns.
Maybe there's an association between them I don't understand?
'begging rector to quit' is the wordplay.
'begging' becomes 'praying' ('pray' can be a synonym of 'beg').
'rector' becomes 'r'.
'to quit' is a deletion indicator.
'praying' with 'r' taken away is 'PAYING'.
Can you help me to learn more?
(Other definitions for paying that I've seen before include "Giving money for good or services" , "Discharging a debt" .)