The Montgomery–Vaughan citation in the forum proof is correct: H. L. Montgomery and R. C. Vaughan, Mean values of character sums, Can. J. Math. 31 (1979), no. 3, 476–487, Theorem 1 states Σ_{χ≠χ₀} M(χ)^{2k} ≪_k φ(q) q^k for any real k > 0, over non-principal characters mod q; with k = 2 this gives the ≪ q³ bound used, with no log factors and no primality assumption on q (the post's spelling 'Vaughen' is a typo).