I've recently started using Lightroom's facial recognition features, which so far have been working very well.
I'm slightly confused about how to reject suggestions when in people view for a specific person though. In the general people view window, there are options to confirm or reject every suggestion. When I look at the suggestions for a specific individual, however, I am only able to confirm suggestions. This means that there are always a handful of incorrect suggestions floating around that I cannot get rid of, making it hard confirm correct suggestions in bulk. FYI, the incorrect suggestions are usually of strangers in the background of photos, so I am unable to simply correct the name. Also, I prefer to not use the general people view window, since it seems to randomise the order of suggestions instead of putting the closest matches first, as is the case when viewing an individual's window.
In the dedicated People view, you can right-click on any of the unwanted face regions in the Unnamed People window and select "Remove Face Region" from the resulting sub-menu.
If in the Loupe view in Library, you can select the "Draw Face Region" icon on the Toolbar. That will show all identified face regions, which can include both already-named faces and unnamed faces. If you want to get rid of any of the identified face regions, click inside the target face region and the region becomes selected and an X in the top-right corner appears. Clicking on the X will delete that face region.