The author proves congruences modulo \(p^2\) and \(p^3\) for the sums \(\sum_{j=1}^{p-1} (np+j)^{-r}\). These generalize results found by \textit{J. W. L. Glaisher} [Q. J. Pure Appl. Math. 31, 321-353 (1900; JFM 31.0185.01)]\ for the case \(n = 0\). The proof uses a \(p\)-adic expansion due to \textit{L. C. Washington} [J. Number Theory 69, 50-61 (1998; Zbl 0910.11047)]\ for the sum \(\sum_{j=1}^{np} \{ j^{-r}: (j,p) = 1\}\) in which the coefficients are \(p\)-adic L-functions. As a corollary, he deduces some generalizations of the classical theorem of Wolstenholme. For example, if \(r\) is odd, \(p \geq r+4\) is an odd prime and \(2n \equiv -1 \pmod p\), then \(\sum_{j=1}^{p-1} (np+j)^{-r} \equiv 0 \pmod {p^3}\).