On some congruence conjectures modulo $p^2$

In this paper, we mainly obtain a congruence which contains a conjecture of Z.-W. Sun. For any prime $p>3$, we have $$ \sum_{n=0}^{p-1}\left(\sum_{k=0}^n\binom{n}k\frac{\binom{2k}k}{2^k}\right)\sum_{k=0}^n\binom{n}k\frac{\binom{2k}k}{(-6)^k}\equiv \left(\frac3p\right)3^{p-1}\pmod{p^2}, $$ where $\left(\frac{\cdot}{p}\right)$ stands for the Legendre symbol.