Infected having caught cold? One might doubt that (7)
I believe the answer is:
sceptic
'one might doubt that' is the definition.
(a sceptic doubts or disbelieves)
'infected having caught cold?' is the wordplay.
'infected' becomes 'septic' (similar in meaning).
'having caught' indicates putting letters inside.
'cold?' becomes 'c' (eg on taps).
'septic' going around 'c' is 'SCEPTIC'.
(Other definitions for sceptic that I've seen before include "One might doubt it" , "Agnostic" , "One doesn't believe" , "Greek philosopher" , "One inclined to question accepted opinions" .)