At home, only once losing head, showing rudeness (9)
I believe the answer is:
insolence
'rudeness' is the definition.
(insolence is a kind of rudeness)
'at home only once losing head' is the wordplay.
'at home' becomes 'in' ('in' can be similar in meaning to 'at home').
'only' becomes 'sole' ('sole' can be a synonym of 'only').
'losing head' means to remove the first letter.
'once' with its first letter taken away is 'nce'.
'in'+'sole'+'nce'='INSOLENCE'
'showing' is the link.
(Other definitions for insolence that I've seen before include "Cheek, irreverence" , "Disrespectful behaviour" , "Lip; cheek" , "Rude behaviour" , "freshness" .)