  We prove various low-energy decomposition results, showing that we can
decompose a finite set $A\subset \mathbb{F}_p$ satisfying $|A|<p^{5/8}$, into
$A = S\sqcup T$ so that, for a non-degenerate quadratic $f\in
\mathbb{F}_p[x,y]$, we have
  \[ |\{(s_1,s_2,s_3,s_4)\in S^4 : s_1 + s_2 = s_3 + s_4\}| \ll |A|^{3 -
\frac15 + \varepsilon}
  \] and
  \[
  |\{(t_1,t_2,t_3,t_4)\in T^4 : f(t_1, t_2) = f(t_3, t_4)\}|\ll |A|^{3 -
\frac15 + \varepsilon}\,.
  \]
  Variations include extending this result to large $A$ and a low-energy
decomposition involving additive energy of images of rational functions. This
gives a quantitative improvement to a result of Roche-Newton, Shparlinski and
Winterhof as well as a generalisation of a result of Rudnev, Shkredov and
Stevens.
  We consider applications to conditional expanders, exponential sum estimates
and the finite field Littlewood problem. In particular, we improve results of
Mirzaei, Swaenepoel and Winterhof and Garcia.
