The distribution of semi-integral points on a class of singular cubic hypersurfaces

Let $k$ be a positive integer and let $X_k$ be the cubic hypersurface defined by the equation $x^3-(y_1^2+\cdots+y_{4k}^2)z=0$. In this paper, we give an asymptotic formula for the counting function of semi-integral points on $X_k$. We also prove that this asymptotic formula agrees with Manin's conjecture for $\mathcal{M}$-points \cite[Conjecture~1.4]{Moe26a} on the $a$-invariant and the $b$-invariant.