Weyl Sums for Quadratic Roots

We bound the Weyl sum$$\sum_{{\scriptstyle b\ (\mathrm{mod}\ c)} \atop {\scriptstyle b^2 \equiv D\ (\mathrm{mod}\ c)}} e\left({hb \over C}\right)$$or, more precisely, sums of this sum averaged over the modulus Y $\sqrt{D}$. This latter fact is crucial in the applications which are our main concern: to the (weighted) equidistribution of real roots of quadratic equations (Linnik–Skubenko problem), to the asymptotic behavior of cycle integrals of the j-function, and to the representability of all large integers in a suitable residue class as the sum of three squares, one being the square of a prime.