How to delete a part of audio in Audition

That looks like a multichannel file - is it? If it is, then open it in Waveform view, select only the line you want (use the track visibility controls at the end of each line to make all of the channels grey except the one you want to alter), highlight the section you want to delete and hit the 'Delete' key. Healing won't delete a section of a file - that's not what it's for at all; it's meant to repair small glitches and clicks.