  As a continuation of our previous papers arXiv:2109.07639 and
arXiv:2202.08196, we study the linearized structure of the manifestly $4D,
\mathcal{N} = 2$ supersymmetric theory of the cubic couplings of the higher
spin gauge superfields to the matter hypermultiplets. We consider in detail the
superfield equations of motion, construct the conserved hypermultiplet
superfield currents, explore their component structure (basically in the
bosonic sector) and compare it with the corresponding currents in the
conventional higher-spin bosonic theory. We thoroughly study the $\mathcal{N} =
2$ spin $\mathbf{2}$ and $\mathbf{3}$ models as instructive examples.
