In the Montgomery-Vaughan's paper ''The exceptional set in Goldbach's problem'', they estimate the following sum:

$$\displaystyle \max_{0<y\leq x}\max_{0<h\leq x} \left(h+\frac{x}{P}\right)^{-1}\bigl|\sum_{y-h<p\leq y}\chi(p)\log p\bigr|$$

supposing that $\chi$ is a primitive non trivial Dirichlet character of modulus $q\leq P=x^{\delta}$, with $\delta>0$. They say that such estimate follow from the analogous estimate given by Gallagher in the paper ''A large sieve density estimate near $\sigma=1$''. This is quite clear, but I wonder for a complete proof because I think that there are trouble with the fact that $y-h$ could be negative and with the dependence of $T$ by $x$ in the use of the explicit formula:

$$\sum_{n\leq x}\chi(n)\Lambda(n)=-\sum_{|Im(\rho)|\leq T}\frac{x^{\rho}}{\rho}+O\left(\frac{x\log^{2}(qx)}{T}\right) \,\,\,\,\,\text{with} \,\, T\leq x^{\frac{3}{4}}$$

Please, could anyone write some details on the estimate of the sum above? In particular, I don't understand why we can arrive to the following form:

$$\sum_{y-h<p\leq y}\chi(p)\log p\ll \sum_{|Im(\rho)|\leq T} y^{\beta-1}\min(y,h) +O\left(\frac{y\log^{2}(qy)}{T}\right)$$

as stated in the Montgomery-Vaughan's paper.