  Our main result gives an adjunction inequality for embedded surfaces in
certain $4$-manifolds with contact boundary under a non-vanishing assumption on
the Bauer--Furuta type invariants. Using this, we give infinitely many knots in
$S^3$ that are not smoothly H-slice (that is, bounding a null-homologous disk)
in many $4$-manifolds but they are topologically H-slice. In particular, we
give such knots in the boundaries of the punctured elliptic surfaces $E(2n)$.
In addition, we give obstructions to codimension-0 orientation-reversing
embedding of weak symplectic fillings with $b_3=0$ into closed symplectic
4-manifolds with $b_1=0$ and $b_2^+\equiv 3$ mod $4$. From here we prove a
Bennequin type inequality for symplectic caps of $(S^3,\xi_{std})$. We also
show that any weakly symplectically fillable $3$-manifold bounds a $4$-manifold
with at least two smooth structures.
