I am a Herrick--my grandfather left Leicester for Canada in 1912--and I have been working for many years on the family's history. I am aware that the family (then Eryicke) was in Leicester about 1480, but I haven't seen any evidence that they were there before. You also say that John Herrick (the prolific gentlemen buried in 1589 at St. Martin's) was a grandson of the aforementioned Thomas. John's father was Thomas (d.1517), but I have not discovered a grandfather by that name. (The parish records for Houghton-on-the-Hill are deficient, and what I know of the family at that time comes largely from wills.)

Kathy subsequently wrote:

the Herrick Family Association has proven recently that the Henry of Salem was not the son of Sir William of Beaumanor--and of course, not every Herrick is descended from that line of the family, including me.

Some useful Genealogy pages can be found at
http://members.bellatlantic.net/~dinabart/genealogy-pages/herrick.htm
http://www.casbah.ac.uk/cats/archive/180/LEICSA001.htm