In a previous paper with the same title [Number Theory, Ulm/FRG 1987, Lect. Notes Math. 1380, 120--136 (1989; Zbl 0674.10032)], the mean value \[ I=\sum_{r=1}^R \int_0^V \biggl| \sum_M^{M'} d(m) g(m,v,y_r) e(f(m,v,y_r))\biggr|^2 \,dv \] was estimated. Here \(d(m)\) is the divisor function, the functions \(f\) and \(g\) satisfy certain conditions, and \(y_r\) runs over a well-spaced set of real numbers. In the present paper, the function \(f\) is supposed to be approximately of the form \(F(x/M)^aB(v,y)\), or of the form \(F(\log x) B(v,y)\), where \(F\) is a parameter, \(a\neq 0,1\), and B(v,y) is bounded. Under this assumption, the previous estimate for I can be refined and simplified. More precisely, the earlier estimate involved the factor \(\min (R^{1/2},(F/M)^{1/2})\) in one term, but now this factor can be removed. Because \(M\ll F\) by assumption, the result is indeed improved. Applications to Dirichlet polynomials and series are given.