sound of sighing, (Cantonese) possessive
嘅 centers on an audible exhalation: the sound of sighing. The Cantonese possessive use is a separate grammatical function that does not share a clear conceptual link with the sighing meaning.
嘅 combines 口 (mouth) with 既, which likely contributes the sound. The character is used to represent a sighing sound, with the mouth radical indicating a vocal expression.
The mouth 口 on the left and 既 on the right suggest a breathy sound escaping the mouth: a sigh.
For カイ, imagine a deep sigh that sounds like a drawn-out 'kai…' as the air leaves your mouth.