Let κ be an U-invariant reproducing kernel and let H(κ) denote the reproducing kernel Hilbert C[z₁, ..., z_d]-module associated with the kernel κ. Let M_z denote the d-tuple of multiplication operators M_z1, ..., M_zd on H(κ). For a positive integer ν and d-tuple T=(T₁, ..., T_d), consider the defect operator

D_{T*, ν}:= ∑_l=0^ν (-1)^l {ν \choose l} {∑_|p|=l(l!/p!){T^p}{T*}^p}. The first main result of this paper describes all U-invariant kernels κ which admit finite rank defect operators D_{M*z, ν}. These are U-invariant polynomial perturbations of R-linear combinations of the kernels κ_ν, where κ_ν(z, w)=(1/(1-ınp{z){w})^ν} for a positive integer ν. We then formulate a notion of pure row ν-hypercontraction, and use it to show that certain row ν-hypercontractions correspond to an A-morphism. This result enables us to obtain an analog of Arveson's Theorem F for graded submodules of H(κ_ν). It turns out that for μ < ν, there are no nonzero graded submodules M of H(κ_ν) (ν ≧ 2) with finite rank defect D_{(Mz|M)*, μ)}.