Hello,
From the symptoms it can be due to contact dermatitis, allergy or eczema. Wash the areas several times with fresh water. Do not use any cosmetic products at the sites. You can apply some calamine lotion at the rash as it will help in soothing the skin. For mild-moderate symptoms a weak steroid may be used, whilst more severe cases require a higher-potency steroid.
Symptoms can be exacerbated by dryness of the skin. Use good quality moisturizers to prevent moisture loss from the skin. During the day, hydrate and protect hands with a protective and moisturizing cream. At night, repair them with creams formulated from glycolic acid—a must for improving skin texture and tone. If still the symptoms do not improve then please get a clinical examination done by a dermatologist. Fungal infections and eczema have to be ruled out then.
Guess it is only an irritation reaction. Application of vaseline should do. If persists a scraping for fungus is done. ok?