I once mentioned this to them in an email and they said that it was something that they currently could not consider due their problems with rolling stock (Dr John Godber) and condition of the Wolds Coast Line, they then went to say that after the upgrade (just taken place- Feb 1/2 term) and the release of the 222's they would begin to look at proposals for extending there services.
Nothing as yet though except Northern are proposing extra services to Scarborough
I'm not sure about demand from Driffield or Bridlington, but I'm certain that there would be demand for a Beverley/Cottingham-London service. I'm certain that local government officials from both East Riding and Hull councils frequently travel to the capital and direct services from Beverley and Cottingham would definitely encourage usage of trains. I'm quite surprised that the local MP hasn't pushed for this, as in the case of Howden, whose service level has improved drastically in recent years as a result of Hull Trains calling there.