The best advice here is to just not buy new. You can get a refurbished laptop from trusted sellers that will definitely exceed the requirements for under $200. If you are willing to spend up to $300 then there is literally no concerns.
For example here is a DELL Latitude 7490 Laptop Intel Core i5 8th Gen 8250U (1.60GHz) 16GB Memory 256 GB SSD Intel UHD Graphics 620 14.0" Windows 10 Pro for $192 on Newegg.
If you don't like Newegg there are tons of other stores that offer refurbished laptops. I assume you are in the USA and stores you can consider are Target, Walmart, Best buy.
There is very little risk with refurbished laptops. They are usually tested before being shipped out and if you buy from a legit company they usually offer some kind of store warranty. Walmart for example offers a 90 day return policy for any reason.
It's hard to make suggestions on which laptop to get since availability plays a bigger role here. So just look around at your desired budget and pick what you think your kid needs.
If you think your kid won't accept a laptop that has a scratch, simply look for "As New". Then there usually aren't any visible damages. But if you are fine with some cosmetic damage you can get real bargains.