The idiom is "handle with kid gloves." It means to treat someone or something very delicately or carefully. The phrase comes from the fact that kid gloves are made from the skin of young goats, which is very soft and delicate.

The phrase "handle with kit gloves" is not correct. It is a misspelling of "kid gloves." The word "kit" is a type of bag or case, and it does not make sense in this context.