Beginning to suppress cough in hovel (5)
I believe the answer is:
shack
'hovel' is the definition.
(I've seen this before)
'beginning to suppress cough' is the wordplay.
'beginning to' says to take the initial letters.
'cough' becomes 'hack' (hacking is a kind of coughing).
The initial letter of 'suppress' is 's'.
's'+'hack'='SHACK'
'in' is the link.
(Other definitions for shack that I've seen before include "Crude dwelling" , "Crudely built hut" , "basic accommodation" , "building made of wood?" , "Rough dwelling" .)