She’s always going for those lozenges - as if this is a typical irritated throat that would be soothed by a lozenge. It’s deception for the audience.
I am sure she has had a few close calls of aspirating that lozenge into her airway, causing suffocation, and that’s why she let it drop into that glass of water.
She knew she wasn’t done coughing, and the optics of Kaine having to perform the Heimlich maneuver on stage wasn’t something she wanted to risk.
Agreed, Hillary prolly is sucking on one or two lozenges to tamp down on the cough impulse, whatever is the source of irratation or “itch”.
Hillary prolly had a spike in cough reflex, and went to the water based medicine.
Hillary prolly did not have the lozenge tucked between her cheek and gum, and panicked thinking the lozenge was going down her windpipe, or to her stomach