We show that in an $m$-step Carnot group, a probability measure with finite $m^{th}$ moment has a well-defined Buser-Karcher center-of-mass, which is a polynomial in the moments of the measure, with respect to exponential coordinates. Using this, we improve the main technical result of our previous paper concerning Sobolev mappings between Carnot groups; as a consequence, a number of rigidity and structural results from that paper hold under weaker assumptions on the Sobolev exponent. We also give applications to quasiregular mappings, extending earlier work in the $2$-step case to general Carnot groups.
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