The Norfolk Coast Path, which is a public footpath, runs along the shore from Wells-next-the-Sea to Burnham Overy Staithe, giving access to the whole of the Holkham Bay beach. If locked gates, etc, are obstructing the free passage of this public footpath, inform the local council's Rights of Way Officer, who should get the obstruction removed.
Did you mean to ask about ways of accessing Holkham Bay beach from inland at intermediate points, rather than from either end at Wells-next-the-Sea or Burnham Overy Staithe? The OS map indicates that the path which links the village of Holkham to the shore at Holkham Gap is part of the National Cycle Network and therefore this should be freely open to both walkers and cyclists. If it is being obstructed, again contact the local council.