Can I use a 2019 desktop Mac screen with a Mac mini?

Sorry but no. Your systems are too new to use Target Display Mode:

Use your iMac as a display with target display mode - Apple Support 

There are third-party solutions such as Luna Display, but none will work if the iMac (which is what I presume you're referring to) isn't working.

Regards.

No, unfortunately. Just put that thought out of your mind.

Plan to buy a proper monitor for a new Mac mini if that's the choice of Mac you make.

If the old iMac is done, recycle it. Drop it at an Apple Store. Staples and Office Max and some other electronics outlets also will recycle old tech.

If possible, do this with the iMac: What to do before you sell, give away, trade in, or recycle your Mac - Apple Support

MacMidget wrote:

My desktop Mac no longer processes, but the screen still works.

What do you mean...? "no longer processes"

If it does not full startup, see > If your Mac doesn't start up all the way - Apple Support

Can I connect a MacMini to it and use it as a monitor?

Same answer as the others. No.

If the iMac is dead, replace it with a new iMac or new Mac mini and external display.