This paper is a continuation of a paper of the first author together with \textit{Y. Yi} [Bull. Lond. Math. Soc. 34, 469--473 (2002; Zbl 1038.11052)]. The main result is a generalized identity of the form \[ \sum_{n=1}^{q}\chi\bigl(f(n)\bigr)= \varepsilon(\chi,f)\cdot q^{1/2} \] where \(q\) is a perfect square, \(\chi\) a primitive Dirichlet character modulo \(q\), \(f\) a certain polynomial with integer coefficients and \(|\varepsilon(\chi,f)|=1\) is explicitly given. As a corollary they got for \(q\) an odd square number and \(m\) any natural number \(m\) with \((q,m)=1\) the following nice identity \[ \sum_{n=1}^{q}\chi\bigl(n^m(1-n)^m\bigr)= \overline{\chi}(4^m)\cdot q^{1/2}. \] For general moduli \(q\), whether there exists a similar formula, is an open problem.