  We study the one-loop effective action for $6D,$ ${\cal N}=(1,0)$
supersymmetric Yang--Mills (SYM) theory with hypermultiplets and $6D,$ ${\cal
N}=(1,1)$ SYM theory as a subclass of the former, using the off-shell
formulation of these theories in $6D,$ ${\cal N}=(1,0)$ harmonic superspace. We
develop the corresponding supergraph technique and apply it to compute the
one-loop divergences in the background field method ensuring the manifest gauge
invariance. We calculate the two-point Green functions of the gauge superfield
and the hypermultiplet, as well as the three-point gauge-hypermultipet Green
function. Using these Green functions and exploiting gauge invariance of the
theory, we find the full set of the off-shell one-loop divergent contributions,
including the logarithmic and power ones. Our results precisely match with
those obtained earlier in [1,2] within the proper time superfield method.
