On sums of powers of consecutive squares over finite fields, and sums of distinct values of polynomials

For each odd prime power q, and each integer k, we determine the sum of the kth powers of all elements [Formula: see text] for which both x and [Formula: see text] are squares in [Formula: see text]. We also solve the analogous problem when one or both of x and [Formula: see text] is a nonsquare. We use these results to determine the sum of the elements of the image set [Formula: see text] for each [Formula: see text] of the form [Formula: see text], which resolves two conjectures by Finch-Smith, Harrington, and Wong.