Improper garment in which Carol's welcomed by father: that's painful! (6,5)

'improper garment in' is the definition.
The definition suggests a singular noun which matches the answer.

'carol's welcomed by father that's painful' is the wordplay.
'carol' becomes 'sing' (carol can mean to sing carols).
'welcomed by' means one lot of letters goes inside another (letters welcomed inside).
'father' becomes 'pop' (pop is a kind of father).
'that's painful' becomes 'ouch' (I've seen this before).
'sing' put inside 'pop' is 'posingp'.
'posingp'+'ouch'='POSING POUCH'

'which' acts as a link.

Can you help me to learn more?