Some congruences involving generalized Bernoulli numbers and Bernoulli polynomials

Let $[x]$ be the integral part of $x$, $n \gt 1$ be a positive integer and $\chi_n$ denote the trivial Dirichlet character modulo $n$. We use an identity established by Z. H. Sun to get congruences of $T_{m,k}(n)=\sum_{i=1}^{[n/m]}\frac{\chi_n(i)}{i^k}$ (mod $n^{r+1})$ for $r=1, 2$, any positive integer $m $ with $n \equiv \pm 1$ (mod $m)$ in terms of Bernoulli polynomials. As an application, we also obtain some new congruences involving binomial coefficients modulo $n^4$ in terms of generalized Bernoulli numbers.