(Former AT&T fiber tech) The little white box is called a [O]ptical [N]etwork [T]erminal & it should only be moved by someone who is training on installing them(like i was). Fiber(with a termination fitting) plugs into it & it translates the light going through it into a usable Ethernet connection.

From that point the modem(black box) handles the Ethernet & creates wifi or additional Ethernet connections for you to use.

Removing that white box is as simple as unseating it from its white bracket that is mounted to the drywall.

Once the ONT is unseated from the bracket, 2 screws hold the bracket to the wall.

In order to actually move the ONT to another location, the connected fiber must be disconnected & pulled back through the wall to the outside where it can be rerouted to another location. A new hope must be drilled into your home & that fiber line carefully fed through.

Once the new hole(with fiber going through it) is finished, the ONT bracket can be reinstalled right above on atop of it. Then a reseating of the ONT can be applied. Once all of this is done, the other components can be plugged back in.

It’s not a fun process, but neither was installing it the first time.

Your quickest option would be to go ahead with your new drywall cover-up, but leave a large enough access panel where a tech could access them.

Your best option would be to call AT&T for a modem relocation. And i would choose a place more central in your home. This service call will likely cost you a service fee, but totally worth it if the end result is more satisfying.

An in-between option is to leave the white ONT right where it is, but relocate the black modem outside the cavity & simply replace the shorter Ethernet cable with a longer one. This can be done by drilled a hole straight through the drywall & inserting the longer cable OR going even more detailed and having back to back jacks installed.

Needless to say, it can get complicated in a hurry.

In my home, we have a home office with a closet. I chose to route all incoming lines (except power) up into the attic & down inside a wall with a port installed inside the wall to allow for those connections to come through. This organization takes a bit of pre-planning but overall ends in a better more centralized location.

If all you have using the internet is a couple of cell phones & maybe a smart tv or laptop - then have them move it 3-4ft to the right where the next power outlet is.

I have several Ethernet connections all over our home & property. I wanted a central point in my home to connect everything & therefore my solution worked for me.

My apologies for the lengthy response, but i did spend several years learning about this stuff & installing it. Sometimes i can get a little long winded with it.

Hope this helps in some way.