The speaker starts with a joke: she has no wish to be remembered for her urine. Dogs, though, like the chowchow and the Welsh corgi, seem to live for it. They sniff their way through the streets, reading the scent messages left by earlier dogs, and add their own, until rain washes the hydrant and tree clean again.
Under the humor is a real longing. The speaker wishes she had some equally simple way to leave herself behind as she walks her neighborhood, something shed from her body, like skin or a tear, that could plant her deeds, her ways, her thinking and her art in the hearts of her friends. The title, "Yearning," points to that desire to be remembered.
The mood is light and playful, but also a little wistful. The comparison is silly on purpose, yet it raises a serious question: how do people leave a lasting mark, when even a dog's mark disappears with the next rain?