  We construct manifestly $4D, \mathcal{N}=2$ supersymmetric and gauge
invariant off-shell cubic couplings of matter hypermultiplets to the higher
integer spin gauge $\mathcal{N}=2$ multiplets introduced in arXiv:2109.07639
[hep-th]. The hypermultiplet is described by an analytic harmonic $4D,
\mathcal{N}=2$ superfield $q^{+}$ with the physical component spins ${\bf s} =
(\frac{1}{2}\,, \;0)$ and an infinite number of auxiliary fields. The cubic
coupling constructed has the schematic structure $q^+ \hat{{\cal H}}^{++}_{(s)}
q^+$, where $\hat{{\cal H}}^{++}_{(s)}$ is a differential analytic operator of
the highest degree $({\bf s} - 1)$ accommodating the massless gauge
$\mathcal{N}=2$ multiplet with the highest spin ${\bf s}$. For odd ${\bf s}$
the gauge group generators and couplings are proportional to ${\rm U}(1)_{PG}$
generator of the internal ${\rm SU}(2)_{PG}$ symmetry of the hypermultiplet and
so do not exist if ${\rm SU}(2)_{PG}$ is unbroken. If this ${\rm U}(1)_{PG}$ is
identified with the central charge of $ 4D, \mathcal{N}=2$ supersymmetry, a
mass for the hypermultiplet is generated and the odd ${\bf s}$ couplings vanish
in the proper massless limit. For even ${\bf s}$ the higher-spin gauge
transformations and cubic superfield couplings can be defined for both massive
and massless (central-charge neutral) hypermultiplets without including ${\rm
U}(1)_{PG}$ generator. All these features directly extend to the case of $n$
hypermultiplets with the maximal internal symmetry ${\rm USp}(2n) \times {\rm
SU}(2)$.
