Name for a bit of firm admonition (4)
I believe the answer is:
irma
'name' is the definition.
'a bit of firm admonition' is the wordplay.
'a bit of' indicates a hidden word (I've seen 'a bit' mean this (the answer is formed by a bit of the clue)).
'IRMA' is hidden within 'firm admonition'.
'for' is the link.
(Other definitions for irma that I've seen before include "Girl's name ("La Douce")" , "The girl for Irish music award" , "woman" , "1963 film with Lemmon and Maclaine" , "Lady" .)