Female rivals at table with the papers? (4)
I believe the answer is:
enid
'female rivals at' is the definition.
The definition suggests a singular noun which matches the answer.
'table with the papers?' is the wordplay.
'table with' becomes 'en' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'the papers?' becomes 'ID' (short for identification papers).
'en'+'id'='ENID'
Can you help me to learn more?
(Other definitions for enid that I've seen before include "Geraint's wife's" , "Miss Blyton" , "Arthurian wife" , "W - - Blyton" , "Geraint's loved one" .)