  We extend the study of corrections to the eikonal approximation that was
initiated in Ref. \cite{Altinoluk:2014oxa} to higher orders. These corrections
associated with the finite width of the target are investigated and the gluon
propagator in background field is calculated at next-to-next-to-eikonal
accuracy. The result is then applied to the single inclusive gluon production
cross section at central rapidities and the light-front helicity asymmetry, in
pA collisions, in order to analyse these observables beyond the eikonal limit.
The next-to-next-to-eikonal corrections to the unpolarized cross section are
non-zero and provide the first corrections to the usual $k_\perp$-factorized
expression. In contrast, the eikonal and next-to-next-to-eikonal contributions
to the helicity asymmetry vanish, while the next-to-eikonal ones are non-zero.
