Right click in my Mac mouse shows up and disappears quickly

Looks like that might be a problem with your mouse. Maybe you have another mouse you can try, or if you have Macbook, try the touch pad, the context menu appears when you click (or tap) the touch pad with two fingers.

You can also open the context menu by holding CTRL and clicking the left mouse button, or holding CTRL and tapping the touch pad with one finger.

If any of that works reliably, the problem is most likely your mouse. You can test changing the batteries, or charging it fully if that is possible, otherwise would probably need to get a new mouse.

With Right-click, the contextual menu is not supposed to go away unless you click...

Which update was it?

Which mouse is this? Does it require its own driver?

If you have access to an alternate mouse, it would be informative to know if it also shows the same problem.

If all else fails give this a try: boot into Safe Mode according to How to use safe mode on your Mac and test to see if the problem persists. Reboot normally and test again.

NOTE 1: Safe Mode boot can take up to 3 - 5 minutes as it's doing the following; 

• Verifies your startup disk and attempts to repair directory issues, if needed

• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)

• Prevents Startup Items and Login Items from opening automatically

• Disables user-installed fonts 

• Deletes font caches, kernel cache, and other system cache files

NOTE 2: if you have a wireless keyboard with rechargeable batteries connect it with its charging cable before booting into Safe Mode. This makes it act as a wired keyboard as will insure a successful boot into Safe Mode.

+1

System Preferences / Setting >> Mouse >> Secondary Click

As long as you do not click Elsewhere - the " scroll-down menu of Copy, Paste, etc etc. " should remain visible