The correct phrase is "handle with kid gloves". Kid gloves refer to gloves made from kid leather, which is the skin of a young goat. Kid leather is soft and smooth, and was once associated with the aristocracy. To "handle with kid gloves" means to handle a situation delicately, and with special tactful consideration, or to be very kind or polite towards someone that you don't want to upset.

"Handle with kit gloves" is a common mistake that some people make, although there is no such phrase as "kit glove" in English.
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