  We study the Markov chain on $\mathbf{F}_p$ obtained by applying a function
$f$ and adding $\pm\gamma$ with equal probability. When $f$ is a linear
function, this is the well-studied Chung--Diaconis--Graham process. We consider
two cases: when $f$ is the extension of a rational function which is bijective,
and when $f(x)=x^2$. In the latter case, the stationary distribution is not
uniform and we characterize it when $p=3\pmod{4}$. In both cases, we give an
almost linear bound on the mixing time, showing that the deterministic function
dramatically speeds up mixing. The proofs involve establishing bounds on
exponential sums over the union of short intervals.
