hello, there are 2 possible reasons for that
one is acrillic caps are made up of synthetic materials. Also perspiration difficult through the cap. Apart from that , the cap make the
scalp warm and sweat more also. Wearing it for longer time make scalp sweaty. so it can produce the worn area a sweat
rash, just like you discribed.
Other thing is it can be,
contact dermatitis to the synthetic material it is made up of.
What can you do is you can change your cap to a wool one which is a natural material and it let perspiration well also. Therefore it cause scalp less sweaty and warm. Also if you are having contact dermatitis to acrillic it would also reduced from wool.
Topical anti steroidal +antibacterial cream would help in healing the rash. keep the area well dried after having bath. Lactocalamine lotion can reduce itching.
Don't use the cap when the climate is warm. it will aggravate the problem.