[For part I cf. J. Lond. Math. Soc., II. Ser. 37, No.3, 385-394 (1988; Zbl 0647.10026).]
Let d,h,r be positive integers and let \(p>r\) be a prime. Consider the set \(S_ 1=\{(m_ 1,...,m_{2r})\in {\mathbb{N}}^{2r};m_ 1+...+m_ r=m_{r+1}+...+m_{2r}\), \(m_ i\leq n\), there exists \(n\in {\mathbb{Z}}\) such that \(p\nmid f_ 1(n)f_ 2(n),\) \(0\equiv F(n)\equiv...\equiv F^{(d)}(n)\not\equiv F^{(d+1)}(n) mod p\}\), where \(f_ 1,f_ 2\), F are given by \[ f_ 1(x)=\prod^{r}_{i=1}(x+m_ i),\quad f_ 2(x)=\prod^{2r}_{i=r+1}(x+m_ i),\quad F(x)=f'_ 1(x)f_ 2(x)- f'_ 2(x)f_ 1(x). \] The main result of the present paper is the upper bound ord \(S_ 1\ll h^{2r-1-d}\), for each \(h\leq p/2r\). Let \(\chi\) be a primitive character mod \(p^{\alpha}\) and let \(\psi\) run through the additive characters mod \(p^{\alpha}\). Then it is possible to derive for some tupel (\(\alpha\),r) and for \(h\leq p^{\alpha /2(r-1)}/2r\) non- trivial estimates for the expressions \[ \sum_{\psi}\sum^{p^{\alpha}}_{n=1}| \sum^{h}_{m=1}\psi (n+m)\chi (n+m)|^{2r}\text{ and } \sum^{N+H}_{n=N+1}\psi (n)\chi (n). \]