Relative number initially cut, that's a weight off my shoulders! (6)
I believe the answer is:
nephew
'relative' is the definition.
(nephew is a kind of relative)
'number initially cut that's a weight off my shoulders' is the wordplay.
'number' becomes 'one' (one is a kind of number).
'initially cut' means to remove the first letter (the initial is cut).
'that's a weight off my shoulders' becomes 'phew' (both are said when relieved).
'one' with its initial letter taken away is 'ne'.
'ne'+'phew'='NEPHEW'
(Other definitions for nephew that I've seen before include "Has a duty (to do something)" , "Sister's son" , "A sibling's child" , "Edward V to Richard III relation" , "Sibling's son" .)