I have only 32 GB free on Macintosh HD and want to make more space by deleting some files.
Many, many files have been copied to an external drive and deleted from Macintosh HD and
the trash has been emptied. I can verify this with the Finder. However Disk Utility still
shows only 32GB free on Macintosh HD. Quitting Disk Utility and restarting doesn’t help
nor does restarting the computer.
Disk Utility needs to display the correct value.
This is important since I have an App that will not
run unless there is more free space on Macintosh HD.
Don't use Disk Utility for this. Instead, go to About This Mac, click on Storage at the top, and then look at the bar showing your internal Mac's hard drive.
Another way to see the same number is to open any Finder window, go to a folder on your drive, like your Home folder, and use View. Show Status Bar. Then you'll see the amount available at the bottom of the Finder window. If you have an iCloud folder selected it will show you iCloud storage, not local storage. That's why selecting something like your Home folder is needed.
These numbers show your real available space. Disk Utility is a system management tool and works at a different level. If the system is using temporary space for things, like memory caches or virtual memory, it doesn't care about that and will report it as being used.
The number you want to worry about for clearing space is the number you see in the Finder or in About This Mac.
Now that said, you never mention the name of this app that "will not run unless there is more free space." So I can't say much more about it other than to speculate that this app could be misleading you about the amount of space it "needs" or perhaps it is miscalculating in some way.
You can contact the developer for support on this. But if you are down to your last 32GB of space on your internal drive than you probably need to clear out some more space anyway as your Mac is probably running slow and you are hitting the ceiling here in other ways.
See https://macmost.com/20-ways-to-free-up-disk-space-on-your-mac.html for some suggestions.