The name pronunciation is Harts-horne; the sh is not a diagraph, as this is a compound.
Hartshorne is undoubtedly an old English place name, derived from the Anglo-Saxon. Its first known appearance is in Domesday Book, about A.D. 1086. The first syllable is from "heort," or "hiort," meaning a stag or buck, while the second is either from "home," or "herne," meaning a meeting place or haunt, or from "horn," meaning a horn. The name may, therefore, have meant either "The Stag's Haunt," or "The Stag's Horn."
Such names were popular among the Saxons, as for instance, "Heorot," the name of the palace of the Danish king, in the epic of Beowulf, so named from the stag's antlers with which it was adorned.