How do I repair the binding of the book where the spine of the cover is too wide?
It looks like a low-budget operation using a one-size-fits-none cover stock, or they got the wrong size covers.  The cover spine is too wide.  As long as it's stiff, it will continue to pull on the page seams.  I would try these two steps:

The original glue location is visible where it came apart.  Put a narrow bead of glue in the same location (Elmer's Glue should be fine).  If you put down too much, spread it from the original location toward the spine and wipe off any excess (when you press it closed, you don't want glue being squeezed between the cover overlay and the front page).
Lightly close the cover and immediately open it again to spread some of the glue to the other side.  If excess glue squeezes onto where you don't want it, wipe that off with a slightly damp rag.  Leave the cover open for a few minutes for the glue to mostly soak into the paper and any wiped areas to dry enough to not be sticky.  Close the book, making sure the cover is aligned as it originally was.  Put a little weight on top, like another heavy book, and let it dry for at least a few hours.

Soften and curve the spine so it's flexible.  The spine will try to pull the pages apart if it's stiff and flat.  Between the spine and the covers, there's an indentation front and back.  Put a heavy duty rubber band in that indentation to compress the covers against the pages.  Then open the book in the middle and lay it flat, pages down.
The spine should jut out in a curve.  If it doesn't, find something rod-shaped that you can push between the spine and the pages to push the spine out into a curve (just big enough to introduce a curve, not so big that it tears the glue joint holding the pages in).  Use your fingers to compress and roll the curve all along the length and from cover to cover to train it into a curved surface instead of a flat one, and to soften it a little.
Then close the book and with the rubber band still on, hold the book spine-down, press it against a surface like a desktop, and roll it from side to side (putting pressure on the front and back corners).  Compress the indented area to make the indentation and the bend between the indentation and the spine more pronounced.
These actions will reduce the tension on the pages so the cover doesn't try to pull the pages apart.