We employ the harmonic superspace methods to study a six-dimensional N=(1,0) supersymmetric gauge theory with higher derivatives coupled to a hypermultiplet in the adjoint representation. By introducing a novel non-minimal interaction between the gauge multiplet and the hypermultiplet, we demonstrate that the one-loop divergences in gauge superfield sector, which are present in the conventional formulation, are canceled. The resulting theory is off-shell one-loop finite in this sector, while preserving the gauge invariance and N=(1,0) supersymmetry. The cancelation mechanism is explicitly verified using both the background field method and the supergraph techniques. Thus, we present an example of the higher-derivative supersymmetric gauge theory in six dimensions which is finite in the vector multiplet sector.