Shut in, having a bit of a cough (3)
I believe the answer is:
hem
'shut' is the definition.
(thesaurus)
'having a bit of a cough' is the wordplay.
'a bit of' suggests taking the first letters.
'a cough' becomes 'em' (I am not sure about this - if you are sure you should believe this answer much more).
The first letter of 'having' is 'h'.
'h'+'em'='HEM'
'in' is the link.
Can you help me to learn more?
(Other definitions for hem that I've seen before include "Sewn border" , "Stitched edge of garment" , "Bottom of a garment" , "Double-backed edge" , "Border; fringe" .)