Let $P=\{1,\dots,n\}$ and $S\subseteq P$. The map $$\nu:\overline{\mathcal{M}}\_{i,S\cup\{q\}}\to \overline{\mathcal{M}}\_{g,P},$$ which attaches to a curve in the domain a pointed genus $g-i$ curve $[D,\{p\_i\}\_{i\in S^c},q]$ at the points labeled by $q$ is usually called a clutching map. The pullback map $$\nu^\*:H^2(\overline{\mathcal{M}}\_{g,P})\to H^2(\overline{\mathcal{M}}\_{i,S\cup\{q\}})$$ is completely described in the paper of Arbarello-Cornalba (<https://arxiv.org/pdf/math/9803001.pdf>) (Lemma 1.4).


In the same paper, it is proven that $$H^2(\overline{\mathcal{M}}\_{g,P}\times \overline{\mathcal{M}}\_{g',P'})\cong H^2(\overline{\mathcal{M}}\_{g,P})\oplus H^2(\overline{\mathcal{M}}\_{g',P'}).$$


What I am wondering is the following:


If we let $S\_1,S\_2\subseteq S$ where $S\_1\cap S\_2=\emptyset$ and consider the "generalized" clutching map $$\mu:\overline{\mathcal{M}}\_{1,S\_1\cup\{q\_1\}}\times \overline{\mathcal{M}}\_{1,S\_2\cup\{q\_2\}}\to \overline{\mathcal{M}}\_{g,P},$$ which attaches a pointed genus $g-2$ curve $[D',\{p\_i\}\_{i\in(S\_1\cup S\_2)^c},q\_1,q\_2]$ at the points $q\_1$ and $q\_2$, can we describe the action of the map $\mu^\*$ on $H^2(\overline{\mathcal{M}}\_{g,P})$?


e.g. what is $\mu^\*(\lambda)$, $\mu^\*(\delta\_{irr})$, etc. in $H^2(\overline{\mathcal{M}}\_{1,S\_1\cup\{q\_1\}})\oplus H^2(\overline{\mathcal{M}}\_{1,S\_2\cup\{q\_2\}})$?


PS: Of course I am interested in the general case not just in moduli space of genus one curves. I just picked it to be as concrete as possible.



"Generalized" clutching maps between moduli spaces of curves
You should read Arbarello-Cornalba more carefully! Let's consider the gluing map
$$ h:\overline M\_{g,n+1} \times \overline M\_{g',n'+1} \to \overline M\_{g+g',n+n'}.$$
 As you say we have $H^2(\overline M\_{g,n+1} \times \overline M\_{g',n'+1}) \cong H^2(\overline M\_{g,n+1}) \oplus H^2(\overline M\_{g',n'+1})$ by the Kunneth theorem, since both factors have vanishing $H^1$. The inclusion of each summand is given by taking the product of a class on one factor with the class of a point on the other factor. But that says that the map $h^\ast : H^2(\overline M\_{g+g',n+n'}) \to H^2(\overline M\_{g,n+1} \times \overline M\_{g',n'+1})$ is the direct sum of the two maps $H^2(\overline M\_{g+g',n+n'})\to H^2(\overline M\_{g,n+1})$ and $H^2(\overline M\_{g+g',n+n'})\to H^2(\overline M\_{g',n'+1})$ given by gluing on any fixed curve (= any point in the moduli space). So the calculations in Arbarello-Cornalba actually determine $h^\ast$ or in fact when you glue any finite product of moduli spaces together, in particular it handles your case which corresponds to gluing three factors of genera $1$, $g-2$ and $1$, respectively.


You can also look at Arbarello-Cornalba-Griffiths "Geometry of algebraic curves, vol II", the section about tautological classes.


