Generating function of some Apéry's series

Question

According to Mathematica, we have $$\sum _{n=1}^{\infty } \frac{H_n }{n^2 \binom{2 n}{n}}z^n=-\text{Li}_3\left(\frac{2}{-z+\sqrt{z-4} \sqrt{z}+2}\right)-\text{Li}_3\left(-\frac{2}{z+\sqrt{z-4} \sqrt{z}-2}\right)+2 i \pi \csc ^{-1}\left(\frac{2}{\sqrt{z}}\right)^2,$$ but how to prove this formula. Moreover, what is the generating function of the apery series $$\sum_{n=1}^\infty \frac{H_n}{n^3\binom{2n}{n}}z^n?$$ More general, $$\sum_{n=1}^\infty \frac{H_n}{n^p\binom{2n}{n}}z^n?\quad (p=0,1,2,3,4,5,...)$$