Judge introducing this writer’s form of jazz (4)
I believe the answer is:
jive
'form of jazz' is the definition.
(I know that jive is a type of jazz)
'judge introducing this writer's' is the wordplay.
'introducing' suggests taking the first letters (I've seen 'introduced' mean this).
'this writer's' becomes 'ive' ('this writer has' means 'I have' for the setter).
The initial letter of 'judge' is 'j'.
'j'+'ive'='JIVE'
(Other definitions for jive that I've seen before include "1930s' big band jazz style" , "Fast American dance" , "Deceptive talk" , "Jazzy dance" , "Style of fast dance music" .)