Let $k$ be an algebraically closed field and $X,Y$ two $k$-schemes. We fix a $k$-point in $X$ and in $Y$ each, which we denote by abuse of notation by $P$. Since the pushout of schemes along closed embeddings exists, we know that there exists a scheme $X\coprod\_{P} Y$. We know that the cocartesian diagram is also cartesian, i.e. it is a Milnor square. My goal is to compute $$H^i\_{ét}(X\coprod\_P Y,\Lambda)$$ where $\Lambda$ is a torsion sheaf coprime to the characteristic of $k$. If we are lead by topological intuition, then we would expect that for $i\geq 0$, we have $H^i\_{ét}(X\coprod\_P Y,\Lambda)=H^i\_{ét}(X,\Lambda)\times H^i\_{ét}(Y,\Lambda).$ The closed embedding of $X\rightarrow X\coprod\_P Y$ yields the long exact sequence
$$H^i\_{X}(X\coprod\_P Y,\Lambda)\rightarrow H^i(X\coprod\_P Y,\Lambda)\rightarrow H^i(Y\backslash P,\Lambda)\rightarrow H^{i+1}\_X(X\coprod\_P Y,\Lambda).$$
Now I probably need to aply excision to understand the relative cohomology, but I'm unable to do it so that I get a sensible result. How can one complete the computation?



Pushout of schemes and étale cohomology
Probably it's better to take a sheaf-theoretic approach. We have maps $z\_X: X \to X\coprod\_P Y$, $z\_Y: Y \to X\coprod\_P Y$, $q: \operatorname{Spec} k \to X\coprod\_P Y$.


There is an exact sequence of sheaves on $X\coprod\_P Y$ $$0 \to \Lambda \to z\_{X\*} \Lambda \oplus z\_{Y\*} \Lambda \to q\_\* \Lambda \to 0 $$ where the arrows are the obvious maps on sections (or arise from the adjunction between $j\_\*$ and $j^\*$), with a minus sign thrown in somewhere, and exactness can be checked on stalks.


This induces a long exact sequence in sheaf cohomology. To get your desired isomorphism, we can use the fact that $H^i ( X\coprod\_P Y, z\_{X\*} \Lambda )= H^i (X, \Lambda)$ because $z\_X$ is a closed immersion, and similarly for all the other maps, and then we only need to check that the natural map $$H^0 ( X\coprod\_P Y , z\_{X\*} \Lambda \oplus z\_{Y\*} \Lambda ) \to H^0 ( X\coprod\_P Y , q\_\* \Lambda ) $$ is surjective, which is easy because we wrote down the map explicitly on sections.


If you want to do it in terms of relative cohomology, you need to consider cohomology relative to the closed subset $Y$, not the open subset $Y \setminus P$, or else you will get junk depending on the geometry of $Y$ near $P$.


