little, polite, slight
The butler gave a little cough to announce his presence.

violent | bad, nasty | chesty, croupy | barking, dry, hacking, racking, rasping | persistent | smoker''s
He had a smoker''s cough and nicotine-yellowed fingers.

give
He gave a slight, apologetic cough and said, ‘Excuse me.’

have, suffer from | catch, develop, get

medicine, mixture, sweet

a bit, a little, slightly | apologetically, discreetly, nervously, politely | loudly, violently | up
He vomited and began coughing up blood.

cough and splutter
The whisky made her cough and splutter.

a coughing fit
He had a coughing fit and couldn''t speak for a few moments.

cough into life
(figurative) The old engine coughed into life.