Suppose I have
$$ e(x) = \left(y - f(x) \right)^2, $$
where $f : \mathbb{R}^n \to \mathbb{R}$ and $y \in \mathbb{R}$. Let's fix $x_0$ and expand $f$ in $x_0$, this yields
$$ e(x) = (y - f(x_0))^2 - 2(y - f(x_0))((\nabla f)(x_0)\Delta x_0 + \Delta x_0^T H(x_0) \Delta x_0) + ((\nabla f)(x_0)\Delta x_0 + \Delta x_0^T H(x_0) \Delta x_0)^2, $$
The term $$((\nabla f)(x_0)\Delta x_0 + \Delta x_0^T H(x_0) \Delta x_0)^2 \approx \left( (\nabla f)(x_0)\Delta x_0\right)^2 = \Delta x_0^T [(\nabla f)(x_0)(\nabla f)(x_0)^T ] \Delta x_0 ,$$ while $$ (y - f(x_0))((\nabla f)(x_0)\Delta x_0 + \Delta x_0^T H(x_0) \Delta x_0) = (y - f(x_0))(\nabla f)(x_0)\Delta x_0 + (y - f(x_0)) \Delta x_0^T H(x_0) \Delta x_0. $$
Therefore we can conclude that
$$ e(x) \approx (y - f(x_0))^2 - (y - f(x_0))(\nabla f)(x_0)\Delta x_0 - (y - f(x_0)) \Delta x_0^T H(x_0) \Delta x_0 + \Delta x_0^T [(\nabla f)(x_0)(\nabla f)(x_0)^T ] \Delta x_0 = (y - f(x_0))^2 - (y - f(x_0))(\nabla f)(x_0)\Delta x_0 - \Delta x_0^T [(y - f(x_0)) H(x_0) + (\nabla f)(x_0)(\nabla f)(x_0)^T ] \Delta x_0 $$
Which means that the hessian matrix of $e(x)$ is given by $$ H_e(x_0) = H(x_0) + (\nabla f)(x_0)(\nabla f)(x_0)^T, $$
however in standard minimization algorithms I've seen many times using $$H_e(x_0) \approx (\nabla f)(x_0)(\nabla f)(x_0)^T,$$
why is the hessian of $f$ entirely rejected?