Kaunakakai is a census-designated place (CDP) in Maui County, Hawaiʻi, United States.It is the largest town on the island of Molokaʻi.The population was 3,425 at the 2010 census. It is  twinned  with Embo in Scotland. The town was made famous in the 1930s by the song The Cockeyed Mayor of Kaunakakai, beginning an ongoing tradition of designating an honorary mayor for the town.t is  twinned  with Embo in Scotland. The town was made famous in the 1930s by the song The Cockeyed Mayor of Kaunakakai, beginning an ongoing tradition of designating an honorary mayor for the town.