On character sums of binary quadratic forms

We establish character sum bounds of the form | ∑ a ⩽ x ⩽ a + H b ⩽ y ⩽ b + H χ ( x 2 + k y 2 ) | p − τ H 2 , where χ is a non-trivial character (mod p), p 1 4 + e H p , and | a | , | b | p e / 2 H . As an application, we obtain that given k ∈ Z ∖ { 0 } , x 2 + k is a quadratic non-residue (mod p) for some 1 ⩽ x p 1 2 e + ϵ .