Question

How do the changing seasons impact Persephone and Demeter's,relationship

Answer 1

Since Demeter was the goddess of the seasons, basically, whenever Persephone was in the Underworld with Hades, it was winter.  There was no autumn or fall in Ancient Greece.  It was winter because Demeter was sad. Once Persephone returned, it would become summer and spring again because Demeter would rejoice.