(To) handle with kid (or kit) gloves

This refers to handling a problem very delicately, or carefully. Kid gloves are made from kid (young goat) or lamb (young sheep), and are very soft and sensitive to the wearer's touch. Kit gloves are work gloves and protect the wearer and the thing, person being handled.

to handle with very gentle care, often to the point of coddling. Kid gloves are very soft leather gloves, typically made from the skin of a young goat (a "kid").