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The answer is conjectured to be $\frac{1}{2}\sqrt{\pi} \sqrt{N}$, and the numerical evidence seems to support this conjecture (see the cited paper of Vaughan and Wooley). The situation is not different for squares versus higher powers for the first moment. The differences appear for fourth moments, since the major arc contribution takes over for squares at the fourth moment, and this has a logarithmic factor.
