It is hard to tell what the spots on the wrist are without seeing them. Given the data you provided (spots on wrist, elevated platelets and blood count-probably meaning WBC-white blood count) it is possible that your son might be experiencing a skin inflammation. More info like: whether the spots multiply and spread affecting a bigger area, whether they hurt and whether they are filled with water-like fluid or pus might have been helpful.
If there is pus it is likely that the spots are caused by a bacterial infection. If it is confirmed by a pediatrician or dermatologist, antibiotics might be recommended. If the spots are recognized to be a sign of an allergy, then additional tests to detect the allergen that caused the allergy might be needed. Once you detect the allergen, avoiding it and taking an antihistaminic might be helpful.
Consult with a pediatrician or a dermatologist about the possible cause for the spots on the wrist and the treatment needed.
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