Our main result gives an adjunction inequality for embedded surfaces in certain -manifolds with contact boundary under a non-vanishing assumption on the Bauer--Furuta type invariants. Using this, we give infinitely many knots in that are not smoothly H-slice (that is, bounding a null-homologous disk) in many -manifolds but they are topologically H-slice. In particular, we give such knots in the boundaries of the punctured elliptic surfaces . In addition, we give obstructions to codimension-0 orientation-reversing embedding of weak symplectic fillings with into closed symplectic 4-manifolds with and mod . From here we prove a Bennequin type inequality for symplectic caps of . We also show that any weakly symplectically fillable -manifold bounds a -manifold with at least two smooth structures.