On series identities involving $\left(\genfrac{}{}{0px}{}{4k}{k}\right)$ and harmonic numbers

The harmonic numbers are those $H_n={\sum }_{0<k\le n}\frac{1}{k}(n=0,1,2,\dots )$. In this paper we confirm over ten conjectural series identities with summands involving the binomial coefficient $\left(\genfrac{}{}{0px}{}{4k}{k}\right)$ and harmonic numbers. For example, we prove the identities $$\sum _{k=1}^{\mathrm{\infty }}\frac{\left(\genfrac{}{}{0px}{}{4k}{k}\right)}{{16}^k}((22k^2-92k+11)H_{4k}-\frac{449k-275}{2}-\frac{85}{12k})=-151-\frac{80}{3}\log 2$$ and $$\sum _{k=0}^{\mathrm{\infty }}\frac{\left(\genfrac{}{}{0px}{}{4k}{k}\right)((11k^2+8k+1)(10H_{4k}-17H_{2k})+2k+18)}{(3k+1)(3k+2){16}^k}=8\log 2,$$ which were previously conjectured by Z.-W. Sun.