Promise to pass the hat round first (4)
I believe the answer is:
oath
'promise' is the definition.
(I know that Oath is a type of promise)
'to pass the hat round first' is the wordplay.
'to pass' means one lot of letters go next to another.
'round' indicates anagramming the letters.
'first' becomes 'o' (I can't justify this - if you can you should give a lot more credence to this answer).
'hat' anagrammed gives 'ath'.
'ath' put after 'o' is 'OATH'.
Can you help me to learn more?
(Other definitions for oath that I've seen before include "Swearword or solemn promise" , "Curse - sworn statement" , "swearing" , "It's taken in court" , "Sworn promise" .)