  The Fourier transform plays a central role in many geometric and
combinatorial problems cast in vector spaces over finite fields. In general,
sets with good uniform bounds for the Fourier transform are less structured,
more `random', and can often be analysed more easily. In many cases obtaining
good \emph{uniform} bounds is not possible, even if `most' points admit good
pointwise bounds. Motivated by this, we propose a more nuanced approach where
one seeks to bound the $L^p$ averages of the Fourier transform instead of only
the maximum. We explore this idea by considering several examples and find that
a rich theory emerges. Further, we provide various applications of this new
approach; including to sumset type problems, the finite fields distance
conjecture, and the problem of counting $k$-simplices inside a given set.
