cackling of fowls to gag, to vomit
嗝 centers on a harsh, involuntary sound or action from the throat: the cackling of fowls, or the gagging and vomiting reflex.
嗝 combines 口 (mouth) with 鬲, which likely contributes the sound. The character is associated with throat-related sounds and actions.
The mouth 口 beside 鬲 suggests a harsh, guttural sound coming from the throat, like a chicken's cackle or a person gagging.
For カク, imagine a chicken cackling loudly: 'kak-kak-kak!' The harsh 'kak' sound cues カク while the chicken cackles.