When building a supersymmetric theory people typically write the superpotential as a products of fields. However, there is another ingredient that could in principle be used - the covariant derivative. Since it still transforms properly under supersymmetry it seems like a perfectly valid operator. Furthermore such terms will be SUSY invariant since we integrate over superspace.
As an example I've seen it written that the most general renormalizable superpotential for a single chargeless field is given by,
W=α+βΦ+mΦ2+λΦ3
However, there could in principle also be a term,
DαΦDαΦ
where
Dα≡∂α+i(σμˉθ)α∂μ. This would give Lagrangian contribution:
L⊃∫d2θDαΦDαΦ+h.c.=F(y)F(y)+h.c.
and since
F is real this would potentially wreck havoc on the scalar potential.
Is there some reason they can be omitted?