On some congruences of certain binomial sums

For any prime (p>3,) we prove that

$$begin{aligned} sum _{k=0}^{p-1}frac{3k+1}{(-8)^k}{2katopwithdelims ()k}^3equiv pleft( frac{-1}{p}right) +p^3E_{p-3}pmod {p^4}, end{aligned}$$

where (E_{0},E_{1},E_{2},ldots ) are Euler numbers and (left( frac{cdot }{p}right) ) is the Legendre symbol. This result confirms a conjecture of Z.-W. Sun. We also re-prove that for any odd prime (p,)

$$begin{aligned} sum _{k=0}^{frac{p-1}{2}}frac{6k+1}{(-512)^k}{2katopwithdelims ()k}^3equiv pleft( frac{-2}{p}right) pmod {p^2} end{aligned}$$

using WZ method.