  In a previous paper we constructed a spectrum-level refinement of Khovanov
homology. This refinement induces stable cohomology operations on Khovanov
homology. In this paper we show that these cohomology operations commute with
cobordism maps on Khovanov homology. As a consequence we obtain a refinement of
Rasmussen's slice genus bound s for each stable cohomology operation. We show
that in the case of the Steenrod square Sq^2 our refinement is strictly
stronger than s.
