Take no notice of Italian man heading off (6)
I believe the answer is:
ignore
'take no notice of' is the definition.
(to take no notice of something is to ignore it)
'italian man heading off' is the wordplay.
'italian man' becomes 'signore' (Italian equivalent to mister).
'heading off' means to remove the first letter.
'signore' with its initial letter taken away is 'IGNORE'.
(Other definitions for ignore that I've seen before include "Forget" , "Region in which to snub" , "Pass over" , "Brush off" , "Cut head off" .)