OP, it's a tricky one. And I'm sorry you are feeling so tired.
Honestly, what you are describing sounds just normal tired l, though, when you have a kid, and due to your life circumstances, rather than your mental health.....if so, then that's what holiday time, or unpaid leave, or going part time is for. It sounds a bit like you are just wanting the company to pay for you to have a break from the life you are in.
Exhaustion does of course cross over into a mental health crisis at some point- if eg you are crying for no reason, or your anxiety is going through the roof, or even when you have slept well, you feel fatigue to the point you can't function, or tasks at work that you know you can normally do, feel overwhelming to you. Once you arrive in that state, or you feel at urgent risk of getting there, then that is real burn out, and I think it is fair to take sick leave. But honestly, what you are saying just sounds a bit more like you don't like your life, hate drop offs and are tired because you are a single mum. None of those are really the company's job to pay for/try to fix.
That said, there is an argument that it's cheaper for the company for you to take a little sick leave now to avoid a real break down later (if that is where you think you are heading). And if you have no other options (no remaining holiday time/ can't afford to take unpaid leave) and it's a huge company with a massive budget, and your colleagues wouldn't be overstrained if you disappeared for a while..... well, it wouldn't be right, but it would be less wrong!
But if you do this, I'd suggest you really use it well to try to get quality rest and reboot..... otherwise, honestly, starting back up to all the things you already hate, will probably feel worse than having never been off. Like walking on blisters.... when you stop for a while, moving again is bloody painful!