Do nursing homes allow their residents to go to work and come back to sleep in nursing home?

An actual nursing home is for people who can't take care of themselves.  These people are not going to have jobs.  You have to be a certain level of "sick" or "elderly" to qualify to be a resident in a nursing home.  

You might be thinking of "assisted living".  Assisted living situations are usually small apartments in one larger building.   In these kinds of homes, you still live fairly independently but someone comes in and helps you with certain things or checks on you from time to time just be be sure you are fine.  In an assisted living situation, you can leave your home at any time for any reason including going to a part time job.  

There are also "senior communities".  These are usually just apartment complexes that only allow "seniors" of a certain age to live there.  (no children are allowed to actually live in the apartments)  In a senior community, there is no one checking on you and you might even still be young enough to be working full time.  Many senior apartment communities start allowing you to live there at the age of 55 and over.

Nursing homes all more freedom to residents who don't have dementia.  

People are in nursing homes because they need skilled nursing care. They aren't in any condition to go to work.

Most of the time, if someone is capable of holding a job they can take care of themselves. So residing in a nursing home would be a moot point.

they might but probably not during covid

People in skilled nursing homes are past the point where they can work. 

But, some people in Assisted Living work, mostly part-time.