  Balog and Wooley have recently proved that any subset $A$ of either real
numbers or of a prime finite field can be decomposed into two parts $U$ and
$V$, one of small additive energy and the other of small multiplicative energy.
In the case of arbitrary finite fields, we obtain an analogue that under some
natural restrictions for a rational function $f$ both the additive energies of
$U$ and $f(V)$ are small. Our method is based on bounds of character sums which
leads to the restriction $\# A > q^{1/2}$ where $q$ is the field size. The
bound is optimal, up to logarithmic factors, when $\# A \geq q^{9/13}$. Using
$f(X)=X^{-1}$ we apply this result to estimate some triple additive and
multiplicative character sums involving three sets with convolutions $ab+ac+bc$
with variables $a,b,c$ running through three arbitrary subsets of a finite
field.
