  We improve higher-order CR Sobolev inequalities on $S^{2n+1}$ under the
vanishing of higher order moments of the volume element. As an application, we
give a new and direct proof of the classification of minimizers of the CR
invariant higher-order Sobolev inequalities. In the same spirit, we prove
almost sharp Sobolev inequalities for GJMS operators to general CR manifolds,
and obtain the existence of minimizers in $C^{2k}(N)$ of higher-order CR
Yamabe-type problems when $Y_k(N)<Y_k(\mathbb{H}^n)$.
