little, polite (esp. BrE), slight
The butler gave a little cough to announce his presence.

violent
bad, nasty, terrible
mild
chesty (BrE)
barking, dry, hacking, racking
chronic, persistent
occasional
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
stifle, suppress
She struggled to stifle a cough.

drop (esp. AmE), medicine, mixture (BrE), remedy (AmE), suppressant, sweet (BrE), syrup

a bit, a little, lightly, slightly
weakly
discreetly, politely (esp. BrE)
awkwardly, nervously, uncomfortably
loudly, violently
harshly
uncontrollably
up
He vomited and began coughing up blood.

cough and splutter
The brandy made her cough and splutter.

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

Cough is used with these nouns as the subject:
engine