Effective equidistribution of unipotent orbits in homogeneous spaces of $\text{SL}(2,\mathbb{R})⋉({\mathbb{R}}^2{)}^k$

Let $G=\text{SL}(2,\mathbb{R})⋉({\mathbb{R}}^2{)}^k$, let $\mathrm{\Gamma }$ be a congruence subgroup of $\text{SL}(2,\mathbb{Z})⋉({\mathbb{Z}}^2{)}^k$, and let $u_{\mathbb{R}}=(u_x{)}_{x\in \mathbb{R}}$ be the one-parameter subgroup of $G$ given by $u_x=(\text{matr}1x01,0)$. We prove polynomially effective asymptotic equidistribution results for expanding translates of $u_{\mathbb{R}}$-orbits and for long pieces of individual $u_{\mathbb{R}}$-orbits in $\mathrm{\Gamma }\mathrm{\backslash }G$. An important ingredient of the proof is the delta symbol version of the circle method.