J.W.
You have created for yourself an impossible schedule. NO ONE can go any length of time with little sleep. Sorry but chances are pretty good you have a perfectly normal son it is you that is the problem.
It is a variation of why do they act up when I can't handle it. Sure it is easy to say that kids have this sixth sense that they know when we can be beat but these are kids, they just don't think like that. They react just as they usually do and then we react differently. It scares the heck out of them and then they say thinks like they hate you.
You need to figure out a way to take care of yourself!! If that means paying a little more to have a few more hours of care so you can sleep then do that!! A four year old does not nap so what you are talking about here is actually very dangerous!! To get any sleep you are having to leave a four year old unattended.
Please, figure out a different way. Eventually your body is going to demand what it needs. You are human, accept those limitations.
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I sincerely wish you lived closer to me. I would watch your son in a heartbeat after work to give you a break. I am not unique! There are people around you that after hearing your story would want to help out. Please reach out! :)