  We construct an off-shell $\mathcal{N}=2$ superconformal cubic vertex for the
hypermultiplet coupled to an arbitrary integer higher spin ${\bf s}$ gauge
$\mathcal{N}=2$ supermultiplet % in flatfour-dimensional space. in a general
$\mathcal{N}=2$ conformal supergravity background. We heavily use
$\mathcal{N}=2, 4D$ harmonic superspace that provides an unconstrained
superfield Lagrangian description. We start with $\mathcal{N}=2$ global
superconformal symmetry transformations of the free hypermultiplet model and
require invariance of the cubic vertices of general form under these
transformations and their gauged version. As a result, we deduce
$\mathcal{N}=2, 4D$ unconstrained analytic superconformal gauge potentials for
an arbitrary integer ${\bf s}$. These are the basic ingredients of the approach
under consideration. We describe the properties of the gauge potentials, derive
the corresponding superconformal and gauge transformation laws, and inspect the
off-shell contents of the thus obtained $\mathcal{N}=2$ superconformal
higher-spin ${\bf s}$ multiplets in the Wess-Zumino gauges. The spin ${\bf s}$
multiplet involves $8(2{\bf s} -1)_B + 8(2{\bf s}-1)_F$ essential off-shell
degrees of freedom. The cubic vertex has the generic structure higher spin
gauge superfields $\times$ hypermultiplet supercurrents. We present the
explicit form of the relevant supercurrents.
