What does the non-zero divergence of $\textbf{H}$-field say about magnetic monopoles?

The short answer is yes. $\textbf{H}$-field can have sources and sinks - these are what the "poles" of a bar magnet are.

In an LIH medium $\textbf{B}=\mu_0\mu_r\textbf{H}$. Even though $\boldsymbol{\nabla}\cdot\textbf{B} = 0$ (always), this does not mean that $\boldsymbol{\nabla}\cdot \textbf{H} =0$ because there is an instantaneous change in $\mu_r$ at the boundaries between media. Lines of $\textbf{H}$-field begin and end on these interfaces, whereas lines of $\textbf{B}$-field are continuous.