  We introduce the notions of pointwise almost h-slant submanifolds and
pointwise almost h-semi-slant submanifolds as a generalization of slant
submanifolds, pointwise slant submanifolds, semi-slant submanifolds, and
pointwise semi-slant submanifolds. We have characterizations and investigate
the integrability of distributions, the conditions for such distributions to be
totally geodesic foliations, the mean curvature vector fields on totally
umbilic submanifolds, the properties of h-slant functions and h-semi-slant
functions, the properties of non-trivial warped product proper pointwise
h-semi-slant submanifolds. We also obtain topological properties on proper
pointwise almost h-slant submanifolds and give an inequality for the squared
norm of the second fundamental form in terms of the warping function and
h-semi-slant functions for a warped product submanifold in a hyperk\"ahler
manifold. Finally, we give some examples of such submanifolds.
