  Let $A(s)$ be a general Dirichlet polynomial and $\Phi$ be a smooth function
supported in $[1,2]$ with mild bounds on its derivatives. New main terms for
the integral $I(\alpha,\beta)=\int_{\mathbb{R}}
\zeta(\frac{1}{2}+\alpha+it)\zeta(\frac{1}{2}+\beta+it)|A(\frac{1}{2}+it)|^2
\Phi(\frac{t}{T})dt$ are given. For the error term, we show that the length of
the Feng mollifier can be increased from $\theta < \frac{17}{33}$ to $\theta <
\frac{6}{11}$ by decomposing the error into Type I and Type II sums and then
studying the resulting sums of Kloosterman sums. As an application, we slightly
increase the proportion of zeros of $\zeta(s)$ on the critical line.
