I don't know about your particular family, but the surnames you mentioned may provide an explanation for the family being called "hawkers" in England. The Irish surname Ó Fionnachta was usually anglicized as Finaghty or Finnerty, and was found mainly in Roscommon, but also in other parts of Connacht. The Irish surname Mac Siúrtáin was often anglicized as Sheridan or Jordan (although some other names were, too). Jordan's and Sheridan's are found all over Ireland, but are/were especially numerous in Mayo and nearby counties. While it's by no means true of everyone named Jordan or Sheridan, those surnames are both found among the "travellers" (known as an lucht siúil in Irish, and sometimes as mincéirí in their own language, Shelta), who were once called "tinkers" (considered a derogatory term nowadays). I don't know whether Finnerty was also a common traveller name. While the travellers lived (and some still live) a gypsy-like life, they were for the most part native Irish people who went on the road centuries ago (there are several theories as to why and how long ago), and are not to be confused with the Romani gypsies, who are of Indian origin. There are now many Irish travellers in Britain, due to emigration, and several communities of them in the US as well.
The travellers were called tinkers, because they went about repairing and trading in metal objects for people (as well as trading horses), and the term "hawkers" used for your relatives might have referred to that. Many have settled down, but a memory of having once been travellers may endure. I have some distant Jordan and Sheridan dna matches myself from Mayo and nearby areas who have vague family lore about some ancestors having once been travellers.
Wikipedia has a good article about the travellers, in case you want to learn more:
https://en.wikipedia.org/wiki/Irish_Travellers
