building on previous work of rozansky and willis, we generalise rasmussen's s-invariant to links in connected sums of $s^1 \times s^2$. such an invariant can be computed by approximating the khovanov-lee complex of a link in $\#^r s^1 \times s^2$ with that of appropriate links in $s^3$. we use the approximation result to compute the s-invariant of a family of links in $s^3$ which seems otherwise inaccessible, and use this computation to deduce an adjunction inequality for null-homologous surfaces in a (punctured) connected sum of $\bar{cp^2}$. this inequality has several consequences: first, the s-invariant of a knot in the three-sphere does not increase under the operation of adding a null-homologous full twist. second, the s-invariant cannot be used to distinguish $s^4$ from homotopy 4-spheres obtained by gluck twist on $s^4$. we also prove a connected sum formula for the s-invariant, improving a previous result of beliakova and wehrli. we define two s-invariants for links in $\#^r s^1 \times s^2$. one of them gives a lower bound to the slice genus in $\natural^r s^1 \times b^3$ and the other one to the slice genus in $\natural^r d^2 \times s^2$ . lastly, we give a combinatorial proof of the slice bennequin inequality in $\#^r s^1 \times s^2$.