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

For each odd prime power [Formula: see text], and each integer [Formula: see text], we determine the sum of the [Formula: see text]-th powers of all elements [Formula: see text] for which both [Formula: see text] and [Formula: see text] are squares in [Formula: see text]. We also solve the analogous problem when one or both of [Formula: see text] 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.