It seems that the following claim is true, but I did not manage to prove it neither to find a reference.

**Claim** Let $f:\mathbb R^p\to\mathbb R$ be a three times differentiable function such that its Hessian is Lipschitz continuous:
$$
\|\nabla^2 f(x)-\nabla^2 f(y)\|\le M\|x-y\|\qquad\forall x,y\in\mathbb R^p,
$$
where the matrix norm is the largest singular value while the vector norm is the usual Euclidean norm. Then, for every $x\in\mathbb R^p$ we have
$$
\|\boldsymbol\Delta [\nabla f(x)]\|^2=\sum_{i=1}^p \bigg(
\sum_{j=1}^p \frac{\partial^3 f}{\partial x_i\partial x_j^2} (x)\bigg)^2 \le pM^2.
$$

I can prove such an inequality with $p^2M^2$ instead of $pM^2$, but do not see how one can remove one power of $p$.