Bill picked up wall game here (4)
I believe the answer is:
eton
'wall game here' is the definition.
(the Eton wall game is played at Eton College)
'bill picked up' is the wordplay.
'bill' becomes 'note' (synonyms).
'picked up' says the letters should be written in reverse.
'note' back-to-front is 'ETON'.
(Other definitions for eton that I've seen before include "College in Berkshire" , "Independent school" , "Berks college" , "Note sent up to English public school" , "Town on the Thames noted for its school" .)