An endomorphism on an algebra $${\mathcal {A}}$$ is said to be strong if it is compatible with every congruence on $${\mathcal {A}}$$ ; and $${\mathcal {A}}$$ is said to have the strong endomorphism kernel property if every congruence on $${\mathcal {A}}$$, other than the universal congruence, is the kernel of a strong endomorphism on $${\mathcal {A}}$$. Here we characterise the structure of Ockham algebras with balanced pseudocomplementation those that have this property via Priestley duality.