  We consider the harmonic superspace formulation of higher-derivative $6D$,
${\cal N}=(1,0)$ supersymmetric gauge theory and its minimal coupling to a
hypermultiplet. In components, the kinetic term for the gauge field in such a
theory involves four space-time derivatives.The theory is quantized in the
framework of the superfield background method ensuring manifest $6D$, ${\cal
N}=(1,0)$ supersymmetry and the classical gauge invariance of the quantum
effective action. We evaluate the superficial degree of divergence and prove it
to be independent of the number of loops. Using the regularization by
dimensional reduction, we find possible counterterms and show that they can be
removed by the coupling constant renormalization for any number of loops, while
the divergences in the hypermultiplet sector are absent at all. Assuming that
the deviation of the gauge-fixing term from that in the Feynman gauge is small,
we explicitly calculate the divergent part of the one-loop effective action in
the lowest order in this deviation. In the approximation considered, the result
is independent of the gauge-fixing parameter and agrees with the earlier
calculation for the theory without a hypermultiplet.
