Can I use toothpicks in power outlet for better grip?
No.

That would not be safe. Absolutely DO NOT DO IT.

In fact, do not even try to stick anything else in the outlet besides the plug.

Now that said, how old is the UPS? Indeed... how old are the batteries in the UPS? Do you need the monitor on a UPS? I suspect if the outlet is frail, then the ups is YEARS old. I'd hypothesize it doesn't even work anymore.

So, go replace the UPS. If the unit has been serviced by you, and you have been replacing batteries, and you absolutely must use the outlet on it in question, replace the outlet. There is a chance you can service the outlet and expand the spring/blades but sincerely, I wouldn't bother.