What is the Sharia ruling for a husband if his wife refuses to fully observe hijab and contents herself with a head covering (tarhah) and modest clothing, and must he do something about her repeated refusal?

You must command your wife to observe the legal hijab if she is not adhering to it. If she insists on abandoning it, then it is preferable to separate from her, unless there are children whose welfare you fear for. This is because there is no good in one who insists on disobeying Allah and does not obey His command. However, if she adheres to the hijab but does not cover her face, the matter is less severe because there is a difference of opinion among jurists regarding this issue. You should explain the evidence to her and encourage her to cover her face with wisdom and good counsel.