An effective equidistribution result for $SL(2,R)⋉(R^2{)}^{\oplus k}$ and application to inhomogeneous quadratic forms

Let $G=$SL$(2,R)⋉(R^2{)}^{\oplus k}$ and let $\mathrm{\Gamma }$ be a congruence subgroup of SL$(2,Z)⋉(Z^2{)}^{\oplus k}$. We prove a polynomially effective asymptotic equidistribution result for special types of unipotent orbits in $\mathrm{\Gamma }\mathrm{\backslash }G$ which project to pieces of closed horocycles in SL$(2,Z)\mathrm{\backslash }$SL$(2,R)$. As an application, we prove an effective quantitative Oppenheim type result for the quadratic form $(m_1-\alpha {)}^2+(m_2-\beta {)}^2-(m_3-\alpha {)}^2-(m_4-\beta {)}^2$, for $(\alpha ,\beta )$ of Diophantine type, following the approach by Marklof [24] using theta sums.