Prompt to back her playing game (the first of many here) (6)
I believe the answer is:
euchre
'game the first of many here' is the definition.
Although both the answer and definition are singular nouns, I can't see how they can define each other.
'prompt to back her playing' is the wordplay.
'prompt' becomes 'cue' (cue can mean a prompt or signal).
'to back' says the letters should be written backwards.
'playing' indicates anagramming the letters.
'cue' reversed gives 'euc'.
'her' is an anagram of 'hre'.
'euc'+'hre'='EUCHRE'
Can you help me to learn more?
(Other definitions for euchre that I've seen before include "Outwit" , "Game of cards" , "American card-game" , "American card game" , "Card game with best, right and left bowers" .)