With a graduate student, I'm going through the paper (Proc. London Math. Soc. (3) 47 (1983), no. 2, 193–224.) We are quite baffled by what follows Lemma 3 on p. 198.
Here's the background and notation. We have two linear forms $l_i(n)=\alpha_in+\beta_i$, $i=1,2$ with $(\alpha_i,\beta_i)=1$, $2|\alpha_i$, as well as several other side conditions ((1.6)-(1.9) p. 194)) which imply that $(l_1(n),l_2(n))=1$.
We have a quadratic character $\chi$ modulo $q$, with Siegel zero $\beta_0$. $\eta=((1-\beta_0)\log q)^{-1}$, so $3\le \eta\ll q$ is known. Let $L=\log q$. We take $$ q^{250}\le x\le q^{500}, $$ and $$ 2\le z\le q, $$ and let $z_0=\log q/\log z\ge 1$. Following the statement of Lemma 3 p. 198 are two consecutive estimates: $$ \sum_{p\ge z}\sum_{x<n\le 2x, p^2|l_i}L^2 2^{\omega(l_1)+\omega(l_2)}\ll xL^2\exp(A L/\log L)z^{-1}, $$ and subsequently $$ xL^2\exp(A L/\log L)z^{-1}\ll xz_0^{-1}, $$ if $z_0\le A\log\log\eta$. Here $A$ is a positive constant which need not be the same at each occurrence.
In the first estimate, we need to read $p^2|l_i$ as $p^2$ divides either $l_1(n)$ or $l_2(n)$ since we already know these are relatively prime. Regardless, I'm completely stumped.
In the second, it would suffice to show $$ L^3\exp(AL/\log L)\ll z\log z. $$ We know $\log\log\eta\ll\log L$ so Heath-Brown's side condition is that $L/\log L\ll z$, but again I am baffled.