  We apply the harmonic superspace approach for calculating the divergent part
of the one-loop effective action of $6D$, ${\cal N}=(1,0)$ supersymmetric
higher-derivative gauge theory with a dimensionless coupling constant. Our
consideration uses the background superfield method allowing to carry out the
analysis of the effective action in a manifestly gauge covariant and ${\cal
N}=(1,0)$ supersymmetric way. We exploit the regularization by dimensional
reduction in which the divergences are absorbed into a renormalization of the
coupling constant. Having the expression for the one-loop divergences, we
calculate the relevant $\beta$-function. Its sign is specified by the overall
sign of the classical action which in higher-derivative theories is not fixed
{\it a priori}. The result agrees with the earlier calculations in the
component approach. The superfield calculation is simpler and provides
possibilities for various generalizations.
