  For fixed coprime polynomials $U,V \in \mathbb{F}_q [T]$ with degrees of
different parities, and a general polynomial $A \in \mathbb{F}_q [T]$, define
the arithmetic function $S_{U,V} (A)$ to be the number of representations of
$A$ of the form $UE^2 + VF^2$ with $E,F \in \mathbb{F}_q [T]$. We study the
mean and variance of $S_{U,V}$ over short intervals in $\mathbb{F}_q [T]$, and
this can be interpreted as the function field analogue of the mean and variance
of lattice points in thin elliptic annuli, where the scaling factor of the
ellipses is rational. Our main result is an asymptotic formula for the variance
even when the length of the interval remains constant relative to the absolute
value of the centre of the interval. In terms of lattice points, this means we
obtain the variance in the so-called ``local'' or ``microscopic'' regime, where
the area of the annulus remains constant relative to the inner radius. We also
obtain asymptotic or exact formulas for almost all other lengths of the
interval, and we see some interesting behaviour at the boundary between short
and long intervals. Our approach is that of additive characters and Hankel
matrices that we employed for the divisor function and a restricted
sum-of-squares function in previous work, and we develop further results on
Hankel matrices in this paper.
