  Let $p>3$ be a prime. In this paper, we obtain the congruences for
$$\sum_{k=0}^{p-1}\frac{w(k)\binom{2k}k^3}{(-8)^k},\
\sum_{k=0}^{p-1}\frac{w(k)\binom{2k}k^2\binom{3k}k}{(-192)^k},\
\sum_{k=0}^{p-1}\frac{w(k)\binom{2k}k^2\binom{4k}{2k}}{(-144)^k}\ \text{and} \
\sum_{k=0}^{p-1}\frac{w(k)\binom{2k}k^2\binom{4k}{2k}}{648^k}$$ modulo $p^2$,
and partial results for $\sum_{k=0}^{(p-1)/2} \binom{2k}k^3\frac{w(k)}{m^k}$
modulo $p^2$, where $m\in\{1,16,-64,256,-512,4096\}$ and
$w(k)\in\{k^2,k^3,\frac 1{k+1},\frac 1{(k+1)^2},\frac 1{(k+1)^3}, \frac
1{2k-1},\frac 1{k+2}\}$.
