supersymmetry - Are there SUSY Lagrangian terms that are not D-term nor F-term?

I've read that a way to construct supersymmetric invariant lagrangian could be either to integrate a superfield in the whole superspace, i.e. in all anticommuting coordinates (D-term), or in half of them (F-term).

Obviously I call F-term a lagrangian term that can't be written as D-term, because all D-term could be written trivially as integrals in half the superspace.

But now I can't understand why couldn't be supersymmetric invariant lagrangian terms that are not even F-term, but they are however invariant.

EDIT

I thought that the answer could be that given an ordinary lagrangian $F(x)$ term (dependent only on space-time coordinates) I can make it a part of a chiral superfields, as a coefficients of $\theta\theta$ in $y-\theta$ expansion

$$ \Phi(y,\theta) = \phi(y) + \sqrt{2} \theta \psi(y) - \theta\theta F(y)$$ $$ y^\mu = x^\mu + i\theta\sigma^\mu\bar{\theta}$$

choosing arbitrarily the $\phi$ and $\psi$ functions. The question now become: does it work?