Decomposition of subsets in finite fields

We extend a bound of Roche-Newton, Shparlinski and Winterhof which says any subset of a finite field can be decomposed into two disjoint subset $\cU$ and $\cV$ of which the additive energy of $\cU$ and $f(\cV)$ are small, for suitably chosen rational functions $f$. We extend the result by proving equivalent results over multiplicative energy and the additive and multiplicative energy hybrids.