cough, sound of laughter
欬 centers on a sudden, forceful expulsion of breath from the throat: a cough, or the explosive sound of laughter that resembles a cough.
欬 combines 亥 (a phonetic element) with 欠 (open mouth, breath), suggesting a sound made with the mouth open, such as a cough or laugh.
The left side 亥 looks like a person doubled over, and the right side 欠 is a wide-open mouth with breath escaping. Picture someone coughing so hard they double over, or laughing with a sudden burst of sound.
For カイ, imagine a loud cough that sounds like 'KAI!' echoing in a quiet room. The sharp 'KAI' sound matches the reading and the sudden cough.