You say "partially black" screen, so part of the display is working? If the phone supports USB OTG then you could connect a mouse to it via an OTG cable and control the phone that way - obviously not while it's connected to the computer, but you may be able to change the default mode (I don't know the LG's menus, it's possible you'd have to look in developer options, which is by default hidden, to change this). Or you may be able to enable another option, such as sharing over WiFi (there are a few apps, such as WiFi File Explorer or Airdroid, which can be used for this). It's really a question of whether you can see enough of the screen to do anything.
Another possibility would be to enable USB debugging, then you could install ADB on your computer (there's a FAQ on it in the FAQs section of the forum) and use that to copy data over USB. You have to be able to unlock the screen to do any of this, but that's where a USB mouse of keyboard comes in.
Most phones of the last few years support OTG, so I'm hoping this will be an option, but it's worth checking the specs of the device.