Beloved kid on one's farm, maybe

PETGOAT

Think of a playful farm animal that might share a name with a cherished companion.

"Beloved kid on one's farm" plays with two meanings of "kid"—it can mean a young goat or a child. The clue hints at a pet (something beloved) that might live on a farm, so combining "pet" with "goat" gives you "PETGOAT." The wordplay is simple: take the literal farm animal and add the affectionate term for a kept animal.