  We prove several supercongruences involving the harmonic number of order two
$H_n^{(2)}:=\sum_{k=1}^n1/k^2$. For example, if $p>5$ is prime and $\alpha$ is
$p$-integral, then we can completely determine $$
\sum_{k=0}^{p-1}\frac{H_k^{(2)}}{k}\cdot\binom{\alpha}{k}\binom{-1-\alpha}{k}\quad\text{and}\quad
\sum_{k=0}^{\frac{p-1}{2}}\frac{H_k^{(2)}}{k}\cdot\binom{\alpha}{k}\binom{-1-\alpha}{k}
$$ modulo $p^3$. In particular, by setting $\alpha=-1/2$, we confirm two
conjectured congruences of Z.-W. Sun.
