You can cut out the segment of your audio where the cough appears (or drop the gain to zero).
But it's not possible to remove the sound of the cough and still leave the speaker's voice up for the half-second or so. You can remove or lower the volume of everything on that audio recording -- but it's not possible to quiet some things and keep others at normal volume on the same recording.
Depending on the exact characteristics of the cough, Adobe Audition's Repair Transient feature might be able to work miracles. I have been amazed at what it can remove, while not affecting the underlying Audio.
If you do Cut out the cough, you might want to find an adequately long section of "silence" which will have some ambient noise in it. Replace the cough with that ambient Audio, so that the viewer is not jarred with complete silence.