This child is a bighead and a miserable specimen (4)
I believe the answer is:
brat
'this child' is the definition.
(brat is a kind of child)
'bighead and a miserable specimen' is the wordplay.
'bighead' becomes 'b' (the head letter of 'big').
'and' says to put letters next to each other.
'a miserable specimen' becomes 'rat' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'b'+'rat'='BRAT'
'is a' acts as a link.
Can you help me to learn more?
(Other definitions for brat that I've seen before include "Brash young type" , "Annoying child" , "An obnoxious or spoiled child" , "Horrible child" , "Marie ---, birth-control pioneer" .)