= Solution
The <ultraviolet softness of the second-order density kernel> requires
$$
F_2(\mathbf q,\mathbf k-\mathbf q)=O(k^2/q^2)
$$
for fixed $\mathbf k$ as $q\to\infty$. In this limit the two hard vectors are opposite,
$$
\frac{\mathbf q\cdot(\mathbf k-\mathbf q)}
{q|\mathbf k-\mathbf q|}\to-1,
\qquad
\frac{|\mathbf k-\mathbf q|}{q}
+\frac q{|\mathbf k-\mathbf q|}\to2.
$$
The constant part of the stated kernel is therefore
$$
\frac57-2A+\frac27=1-2A.
$$
It must vanish, giving
$$
\boxed{A=\frac12}.
$$
This recovers the standard <standard perturbation theory density kernel> $F_2$.
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