First of those children to produce a Kleenex (6)
I believe the answer is:
tissue
'a kleenex' is the definition.
('tissue' can be a synonym of 'kleenex')
'first of those children' is the wordplay.
'first of' suggests taking the first letters.
'children' becomes 'issue' (I've seen this before).
The first letter of 'those' is 't'.
't'+'issue'='TISSUE'
'to produce' acts as a link.
(Other definitions for tissue that I've seen before include "full of lies?" , "Cells sharing a function" , "Fine weave" , "Paper hankie" , "Thin paper; concoction (of lies)" .)