After a month, Fitzgerald's written shortish book (7)
I believe the answer is:
novella
'written shortish book' is the definition.
'novella' can be an answer for 'book' (I've seen this before). I am unsure of the 'written shortish' bit.
'after a month fitzgerald's' is the wordplay.
'after' says to put letters next to each other.
'a month' becomes 'Nov' (abbreviation for November).
'fitzgerald' becomes 'Ella' (musician Ella Fitzgerald).
'nov' put next to 'ella' is 'NOVELLA'.
Can you help me to learn more?
(Other definitions for novella that I've seen before include "story, short one" , "short book" , "Short fictional piece" , "Short work of fiction" , "Slim work of fiction" .)