I had to fire someone like that. Very unproductive employee, started her shenanigans on DAY ONE, it turned out. I was stunned at the time.
But I'm wiser, now. She seemed to arrive on the job "pre-trained" to try to play this kind of game.
And yes (for the curious minded) she could have been Obama's daughter. Very sad...when given a chance (good job, very secure public sector, good pay, health/vacation benefits) she chose to throw all of it away.
As if she came from another country, frankly. Different mores, different habits, different way of thinking. A very short-term mode of thought.
And YES, if you're a competent and caring manager, you CAN fire public sector employees. It's a myth that you can't...at least in Nevada. (I don't know about NYC/Chicago/NJ.)
I have had about a dozen-and-a-half EEOC complaints lodged against me for firing or disiplining people. Nine of them were from one woman. She always had at least three of them going at a time so that she could not be fired until they were resolved. As soon as one was nearing a finding, she would file a new one.
Anyway, I won every single one of them. That was important. When EEOC says I did not do anything wrong, the person has to pay a lawyer to file a lawsuit. If the EEOC ever said I was wrong, he or she could get a lawyer on contingency.