Greetings, Kaye!Do you by chance know which island your father and his family were from?Or the names of his parents?Or his exact birth date?That would all help to narrow it down.
There was a William Hiram Cates from Marsh Harbour, Abaco who married Vernie Roberts.In 1924, Vernie's baby (perhaps not with William?) died of measles, and William died shortly after.Was your father perhaps William Hiram Cates, Jr.?Not sure which marriage he would be from in that case.
There was also a Dennis Cates from Great Guana Cay, Abaco who was the father of Greenwood, Murray, and James Cates.That sounds like it might be the right family as well, though it's hard to tell without dates.It's typical for Bahamians to name their children after relatives, so it could be that your Uncle Dennis had these sons named Greenwood, Murray, and James.
My understanding is that many from the Kates/Cates family emigrated to Key West and beyond around the end of the 1800s and the start of the 1900s, so I'm not sure how many are left in the Bahamas.It's not a family that comes up often in Bahamian genealogy, unfortunately.
If you can gather up more precise information about him, like his birth date and his parents' names (if you have them), the National Archives could do a search for you for a low cost.