Looks more like eczema than a reaction to something it would have gone by now. Have you tried child's farm cream ?everybody is saying it's helping children's eczema might be worth a try

Get some blood tests done to see if he alergic to any types of foods. My son has slight alergy to eggs and dairy which gives him flair ups. Also keep an eye on his face after eating differnt types of food to see if it gets worse to keep track as i found my sons face gets rashy when he has tomato sauce even tho it didnt show up on alergy test

Also I second child's farm. Cheap, gentle and very effective - I actually use it as my daily moisturizer.

Maybe dairy products? Washing powder? Soap? Keep ph neutral. Also water might be an issue. Soft water helps....

My son was the same I found the oilatum cream good to clear up the spots and dryness keeping him moisturised is the trick also I found when we were on holidays the sun cleared it up completely which was fantastic I'm currently using Aveeno Dermexa body wash and cream its early days but so far so good