Ditch the cotton! Cotton sucks. You want a good synthetic, or merino wool.
Cotton may be breathable in the sense that it lets air through, but what you really want is a fabric which wicks moisture and sweat away: merino wool and synthetic fibres do, i.e., when your body sweats, they help "transport" sweat away from your body, and they do not retain humidity.
Cotton, by contrast, soaks up humidity and remains wet. That's the exact opposite of what you want when you sweat!
I swear I will never understand why people are so obsessed with cotton; especially for the summer - there are still people who describe cotton as good for the summer, which beggars belief.
The feel on the body is a matter of subjective opinion, but the properties of the fabrics are not.
E.g. how many athletes do yo see playing with cotton t-shirts?
Or how many hikers hiking with anything made out of cotton?
Also, the manufacturers of some specialist (think serious hiking or motocycling) boots with waterproof membranes like GoreTex specifically state that you shouldn't wear cotton socks for this very reason.
Decathlon has good synthetic socks for kids at reasonable prices.
Ecco has bamboo socks for kids - good but more expensive.
You can also find merino socks for kids (icebreaker, smartwool, etc) but they tend to be expensive and thicker, so better suited to the winter than to the summer (I don't mind lightweight merino in the summer but it's subjective).
Many people dislike synthetic fibres because they think they smell, but that's not always true, it depends on the fibre. A good synthetic can smell less than cotton.
The one problem of many synthetic fibres is the environmental impact because some shed microplastics. Merino doesn't, of course.