  We present results for one-loop matching factors of four-fermion operators
composed of HYP-smeared staggered fermions. We generalize previous calculations
by using the tree-level improved Symanzik gauge action. These results are
needed for our companion numerical calculation of $B_K$ and related matrix
elements. We find that the impact on one-loop matching factors of using the
improved gluon action is much smaller than that from the use of either HYP
smearing or mean-field improvement. The one-loop coefficients for mean-field
improved, HYP-smeared operators with the Symanzik gauge action have a maximum
magnitude of $O(1)\times \alpha_s$, indicating that perturbation theory is
reasonably convergent.
