Minor injury from bit of egg in pop gun? (4,3)

'minor injury from bit' is the definition.
I know nothing about this answer so I cannot judge whether this works.

'of egg in pop gun?' is the wordplay.
'of egg' becomes 'e' (I can't justify this - if you can you should give a lot more credence to this answer).
'in' indicates putting letters inside.
'pop' becomes 'dad' (both can mean one's father).
'gun?' becomes 'arm' (gun is a kind of arm).
'e' placed into 'dad' is 'dead'.
'dead'+'arm'='DEAD ARM'

Can you help me to learn more?