What Ate means
Ate is a Mexican sweet name of Spanish origin meaning "fruit paste". It comes from Spanish -ate (suffix meaning 'paste made of'), as in membrillate, guayabate. It is pronounced AH-teh, and it is most associated with Michoacán.
A rare case of a suffix becoming a word. Spanish had membrillate (quince paste) and guayabate (guava paste), from Catalan codonyat, and Mexican speakers cut the fruit off the front and kept -ate as a noun.
This is a site about what Mexican words mean, so there is no recipe here. For the name itself: From colonial convent sweets like ate and jamoncillo to the chili-and-tamarind candies of the modern corner shop. Alegría means "joy", borrachito means "little drunk one", and muégano's name is a mystery even to Mexicans.
