  Let $n$ be a positive multiple of $4$. We establish an asymptotic formula for
the number of rational points of bounded height on singular cubic hypersurfaces
$S_n$ defined by $$ x^3=(y_1^2 + \cdots + y_n^2)z . $$ This result is new in
two aspects: first, it can be viewed as a modest start on the study of density
of rational points on those singular cubic hypersurfaces which are not covered
by the classical theorems of Davenport or Heath-Brown; second, it proves
Manin's conjecture for singular cubic hypersurfaces $S_n$ defined above.
