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Let (X, τ ) and (Y, σ) be topological spaces, and let τh(X) and σh(Y ) denote the associated topologies formed by the h-open subsets. We study weakly h-continuous functions, defined by requiring the image of a suitable h-open neighbourhood in the domain to lie inside the h-closure of each ordinary open neighbourhood of the corresponding image point. The associated-topology viewpoint shows that this is a genuinely mixed notion: weak h-continuity implies classical weak continuity from (X, τh(X)) to (Y, σ), but the converse need not hold unless the codomain is h-fixed. We identify weak h-continuity as the specialization of the (i, j)-weakly m-continuous framework of Noiri and Popa obtained by taking the minimal structure mX = τh(X) and the codomain bitopology (σ, σh(Y )). We record the corresponding characterizations in h-notation, give explicit finite examples, and establish h-specific composition, retraction and separation results. We introduce weakly h-irresolute functions, which are exactly weakly continuous functions between the associated spaces, and derive corresponding preservation properties. Closed-graph and graphmap results are formulated with the necessary distinction between the associated topology of a product and the product of the associated topologies. Finally, we clarify the role of relative h-compactness and formulate the classical minimal-structure finite-cover and frontier results in the present h-setting.