Greek Myth About Mortal Horse Hitched Beside Immortal Horse

Someone described a story from greek mythology where for some reason one of the immortal horses that pulls some chariot or something (like the sun across the sky every day maybe?) is out of commission or something, so they have to hitch a mortal horse to the chariot to replace him. The mortal horse is able to pull its weight, the mission is accomplished, but at the end the mortal horse dies from the effort.

I'm trying to discover the name of the horse and the story, I want to read it in full. Thanks very much for anyone who can recall the source.