An edgy little cough? (4)
I believe the answer is:
ahem
'little cough?' is the definition.
The definition and answer are not the same part of speech.
'an edgy' is the wordplay.
'an' becomes 'a'.
'edgy' becomes 'hem' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'a'+'hem'='AHEM'
Can you help me to learn more?
(Other definitions for ahem that I've seen before include "Sound made when clearing throat" , "Sound used to attract attention" , "Throat clearing sound to catch attention" , "Expression of disapproval" , "Excuse me!" .)