Can apple back up my phone with a broken screen?

Nobody can backup an iPhone if they cannot interact with its software via the touch screen. That’s like trying to use a computer without a keyboard or mouse - there is no way to select or execute commands on a touch screen device with no functional touchscreen.

As others have mentioned, neither you or Apple can back up your iPhone without a functioning touchscreen

Are you trying to back up your iPhone to restore it on a replacement or do you plan on upgrading?

If you were planning on replacing the damaged iPhone, you can have your touchscreen replaced at a third party repair center

No

You supposed to have periodically backup to iCloud, or iTunes in personal computer before iphone gets busted. So what other post wrote: