Telegraph's first edition paper (6)
I believe the answer is:
tissue
'paper' is the definition.
(tissue paper is a kind of paper)
'telegraph's first edition' is the wordplay.
'first' indicates taking the first letters.
'edition' becomes 'issue' (I've seen this before).
The first letter of 'telegraphs' is 't'.
't'+'issue'='TISSUE'
(Other definitions for tissue that I've seen before include "Thin paper; concoction (of lies)" , "Absorbent paper, used as handkerchief, say" , "Disposable handkerchief" , "Type of paper" , "Thin stuff" .)