Thank you, Osprey, for that suggestion.
Have just looked on the history of parliament site, which tells me that the first mayor & deputy were James Trewynnard & Henry Pyne, in 1529, and the mayors/deputies for 1536, 1539 and 1542 are unknown. So, maybe there was a John Hunkin amongst them (I know this family lived in Liskeard and were landowners there).
I also know that a later John Hunkin marries the daughter of a later mayor of Liskeard, Jane Connock. The records seem scrappy for the earlier 1500s.
Thanks again for your help,
Frosty