The icon is a white rectangle with the word “Mobile…” and a folder symbol. When I tap the icon, the folder becomes an arrow pointing left.
Sorry no one has answered, it seems no one recognizes the icon from the description. If you’re still pondering it, could you tell us which phone you have, and perhaps post a screenshot so we can see the icon?
Here’s the screen shot. I have a Moto X. First generation, I think, but I don’t remember how to check that.
Looks like it is a bad widget on the home page. Probably the app removed the widget in its latest incarnation or you removed whatever app it was for. I would just long-press it and throw it in the trash.