Heading off one that was close relative (6)
I believe the answer is:
nephew
'relative' is the definition.
(type of relative)
'heading off one that was close' is the wordplay.
'heading off' means to remove the first letter.
'that was close' becomes 'phew' (both are said after narrowly avoiding danger).
'one' with its initial letter removed is 'ne'.
'ne'+'phew'='NEPHEW'
(Other definitions for nephew that I've seen before include "Family relative" , "Has a duty (to do something)" , "A young male relation" , "Sister's son" , "Kinsman" .)