We study the function field analogue of a classical problem in analytic number theory on the sums of the generalized divisor function in short intervals, in the limit as the degrees of the polynomials go to infinity. As a corollary, we calculate arbitrarily many moments of a certain family of L-functions, in the limit as the conductor goes to infinity. This is done by showing a cohomology vanishing result using a general bound due to Katz and some elementary calculations with polynomials. This method is based on work of Hast and Matei, except that thanks to a trick involving the logarithmic derivative, we are able to achieve a much smaller error term than is possible by this method for a "typical" problem of this type.