Rational points on cubic hypersurfaces that split off two forms

We show that if $X\subseteq {\mathbb{P}}^{n-1}$, defined over $\mathbb{Q}$ by a cubic form that splits off two forms, with $n\ge 11$, then $X(\mathbb{Q})$ is non-empty. The same holds for an $(m_1,m_2)$-form with $m_1\ge 4$ and $m_2\geq 5$.