What does it mean when it says, "People will finish updating when Photos is in the background." I'm trying to add a person to "People," but haven't figured out how to do that.

New people get added when there is a new person to add (not being glib, really...)

So you have to find a picture of "Mary Ann" that says "unnamed" under it, and type in her name. You can type Mary, or Mary Ann, or Mary Ann Smith, or "granddaughter #1"-- anything you like. You can change it later. Then that picture with that name will show up in the people list. If Photos finds others who look like that, it will put them in that slot, too.

Use the People album in Photos on your iPhone, iPad, or iPod touch - Apple Support

I want to comment on "If Photos finds others who look like that,"

Like it or not, all babies look kind of alike. Photos will have a hard time. Another person here,

léonie, has some suggestions about how to deal with Photos asking you if this picture is of Mary Ann.

Show "Confirm Additional Faces" photos de… - Apple Community

Also, add a keyword, maybe MaryAnn, to the Photos with her in them. Keywords export with pictures, and they can be read by other viewers. Face recognition only works in Photos.

Glad it helped