  For a general renormalizable ${\cal N}=1$ supersymmetric gauge theory with a
simple gauge group we verify the ultraviolet (UV) finiteness of the two-loop
matter contribution to the triple gauge-ghost vertices. These vertices have one
leg of the quantum gauge superfield and two legs corresponding to the
Faddeev--Popov ghost and antighost. By an explicit calculation made with the
help of the higher covariant derivative regularization we demonstrate that the
sum of the corresponding two-loop supergraphs containing a matter loop is not
UV divergent in the case of using a general $\xi$-gauge. In the considered
approximation this result confirms the recently proved theorem that the triple
gauge-ghost vertices are UV finite in all orders, which is an important
ingredient of the all-loop perturbative derivation of the NSVZ relation.
