  We derive $p$-adic expansions for the generalized Harmonic numbers
$H^{(j)}_{p-1}$ and $H^{(j)}_{\frac{p-1}{2}}$ involving the Bernoulli numbers
$B_j$ and the the base-2 Fermat quotient $q_p$. While most of our results are
not new, we obtain them elementarily, without resorting to the theory of
$p$-adic L-functions as was the case previously. Moreover, we show that
\begin{equation*}\sum_{j=0}^{n-1}\left(\frac{(2^{j+1}-1)}{(j+1)}\frac{(2^{j+2}-1)}{(j+2)}\frac{B_{j+2}}{2^{j}}H^{(j+1)}_{\frac{p-1}{2}}+2(-1)^j\frac{q_p^{j+1}}{j+1}\right)p^j\equiv
0 \pmod {p^n} \end{equation*} holds under the condition that $p
>\frac{n+1}{2}$. This is another generalization, modulo any prime power, of the
old $p$-congruence $H_{\frac{p-1}{2}}+2q_p \equiv 0 \bmod p$ attributed to
Eisenstein, which is stronger than the one which has been published recently.
