mollycoddle (n./v.)

because of "molly" (from the woman's name) but even more because the noun is defined as "a pampered, spoiled, effeminate boy or man," this word is thoroughly sexist. For the noun, use spoiled/immature/irresponsible person, timid/gentle/sensitive person. For the verb, use indulge, spoil, pamper, dote on, overprotect. See also mama's boy, sissy.