Series with summands involving harmonic numbers

For each positive integer $m$, the $m$th order harmonic numbers are given by $$H_n^{(m)}=\sum _{0<k\le n}\frac{1}{k^m}(n=0,1,2,\dots )\mathrm{.}$$ We discover exact values of some series involving harmonic numbers of order not exceeding four. For example, we conjecture that $$\sum _{k=0}^{\mathrm{\infty }}(6k+1)\frac{{\left(\genfrac{}{}{0px}{}{2k}{k}\right)}^3}{{256}^k}(H_{2k}^{(3)}-\frac{7}{64}{H}_k^{(3)})=\frac{25\zeta (3)}{8\pi }-G,$$ where $G$ denotes the Catalan constant ${\sum }_{k=0}^{\mathrm{\infty }}(-1{)}^k\mathrm{/}(2k+1{)}^2$. This paper contains $70$ conjectures posed by the author during 2022–2023.