Singing videos, do you already have some kind of mic setup? If not you sould def. put part of your budget towards that and part towards a new camera.
What is the tablet not doing what you would like the new camera to do?
I do want to agree with @Yilk81 that investing in sound first is a good move for singers. You can make poor video look better with lighting and positioning, but you can't easily fix poor sound.
If you have a computer you can record to, I recommend starting with a good webcam. I recommend the AverMedia PW513. It is a very high-quality webcam, supports 4k, etc. With decent lighting it will serve you very well and at only $250US it will be easy on your budget.
I recommend that because the next tier would really be to go to a DSLR and those are out of your price range. You would be better off with the AverMedia and saving towards something like a Canon M50 or a Sony A6X00 camera over spending everything on a camera that won't have as good of quality as a webcam and won't be as versatile as a DSLR.
That said, if you are limited to a tablet, you may want to consider a small camcorder that allows you to record directly to the camera. If that's the case, the Canon Vixia R800 gets recommended a lot and would be in your budget range.
You hear people say this often: Sound is half of the experience. I would argue that it's actually slightly more than that. I would sit through slightly bad video quality but i will NOT sit through bad sound.
I just got myself a Canon M50. It's a $600 range camera. I have ideas for my videos that required a camera that was not connected to my PC like my Logitech C920. I needed to be able to record myself separate from my gameplay plus I wanted higher quality videos as well. For your budget and dependent on what exactly you're gonna record with it, I think a webcam like the Logitech C922x, C925-e or even the BRIO can provide the quality video you may be looking for within your budget. I tried finding cameras within your budget range and couldn't find one worth getting.
I would actually argue that the iPad might have better quality then the logitech webcams. Depending on what gen. iPad it is. And it is more portable then a laptop/desktop computer with a webcam attached.
I film on a six year old Fujifilm camera and the quality is pretty dang good. The camera itself can be had for €200,- used without a lens. It's not for a total newbie but if you have the knowledge to work with it the results are really good. Check out my vids to see the quality. Actually, check out my send video as I got something wrong in my first video and shot in 720p in stead of 1080p
For singing videos, you definitely want to get your sound studio set up first. You'd almost be better off buying a Focusrite Scarlet Solo or 2i2. Get the mic you want, and make singing videos. Even a Sure SM58 would be light years better than any webcam mic.
After that I'd save for a Canon M50 and call it a day. Don't forget: You can get a used Panasonic GH4 for about $400 that will do the job just fine. Used gear can be a good way to go.