"warm covering for the hands," 1590s, from Dutch mof "a muff," shortened from Middle Dutch moffel "mitten, muff," from Middle French moufle "mitten," from Old French mofle "thick glove, large mitten, handcuffs" (9c.), from Medieval Latin muffula "a muff," of unknown origin. In 17c.-18c. also worn by men. Meaning "vulva and pubic hair" is from 1690s; muff-diver "one who performs cunnilingus" is from 1935.
"to bungle," 1827, pugilism slang, probably related to muff (n.) "awkward person" (1837), perhaps from muff (n.) on notion of someone clumsy because his hands are in a muff. Related: Muffed; muffing.
To fail; botch, esp by clumsiness •The older example refers to playing cricket: This is a ripe one. Don't muff it, Billy (1837+)
[verb sense fr the clumsiness of someone wearing a muff on the hands]