One fighting to grab a pencil's lead — it can be used for doodling (5,5)
I believe the answer is:
scrap paper
'it can be used for doodling' is the definition.
(paper which might be used for doodling)
'one fighting to grab a pencil's lead' is the wordplay.
'one fighting' becomes 'scrapper' (someone getting into scraps).
'to grab' indicates putting letters inside.
'lead' indicates taking the first letters.
The initial letter of 'pencils' is 'p'.
'a'+'p'='ap'
'scrapper' enclosing 'ap' is 'SCRAP PAPER'.
(Other definitions for scrap paper that I've seen before include "Make jottings on this!" , "What shamefaced News International decided to do" , "somewhere to make notes" .)