  We compute next to eikonal (NE) and next to next to eikonal (NNE) corrections
to the Lipatov vertex due to a finite target thickness. These arise from
electric field insertions into the eikonal Wilson lines. We then derive a
$k_T$-factorization formula for single inclusive gluon production at NNE
accuracy and find that nuclear effects are absent. We also analyze NNE
corrections to two-gluon production where some of the contributions are found
to exhibit corrections proportional to $A^{2/3}$.
