Kinsman skirting Tyneside with huge sigh of relief (5-6)
I believe the answer is:
'kinsman' is the definition.
(great-nephew is a kind of kinsman)
'skirting tyneside with huge sigh of relief' is the wordplay.
'skirting' means one lot of letters goes inside another.
'tyneside' becomes 'ne' (region in North East of England).
'with' is a charade indicator (letters next to each other).
'huge' becomes 'great' (I've seen this before).
'sigh of relief' becomes 'phew' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'great'+'phew' is 'greatphew'.
'ne' going inside 'greatphew' is 'GREAT-NEPHEW'.
(Other definitions for great-nephew that I've seen before include "Male blood relative" , "Wife's sister's grandson" .)