We study extensions of piecewise polynomial data prescribed on faces and possibly in elements of a patch of simplices sharing a vertex. In the $H^1$ setting, we look for functions whose jumps across the faces are prescribed, whereas in the H(div) setting, the normal component jumps and the piecewise divergence are prescribed. We show stability in the sense that the minimizers over piecewise polynomial spaces of the same degree as the data are subordinate in the broken energy norm to the minimizers over the whole broken $H^1$ and H(div) spaces. Our proofs are constructive and yield constants independent of the polynomial degree. One particular application of these results is in a posteriori error analysis, where the present results justify polynomial-degree-robust efficiency of potential and flux reconstructions.