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The moments of exponential sums which are restricted to a certain subset of the integers can reveal interesting additive information about these sets. With this in mind, Bruedern, Granville, Perelli, Vaughan, and Wooley examined the first moment of exponential sums over the r-free integers, which are integers not divisible by r-th powers. Later, Keil extended this to a result giving the exact order of magnitude of the k-th moments for all k > 0, with the exception of the case k = 1 + 1/r, where he missed by a factor of log(X). We examine the analogous problem in the function field setting, obtaining a result analogous to Keil's, with two added improvements. In the case k = 1 + 1/r, we obtain the exact order of magnitude, and in the case k > 1 + 1/r, we refine the result to an asymptotic formula using the Hardy-Littlewood circle method. We also show that each of these refinements may be adapted to the integer setting, to a weaker effect.