He is best known for his The Progress of Human Culture, a series of six historical and allegorical paintings for the Royal Society of Arts in London.
Edmund Burke was an acquaintance and patron of Barry who funded his travels to Paris, Rome, Florence, and Bologna to study renaissance artwork. Upon Barry’s homecoming, Burke and Reynolds became close friends, causing Barry to become jealous. Perhaps Blake is speaking about this fracture in the relationship between artist and patron.
According to Andrew Graham-Dixon, Blake intended to write a six book epic-poem entitled “James Barry” (130). For more information, reference Graham-Dixon's A History of British Art.