Atopic Dermatitis (Eczema): This condition is characterized by itchy, scaly patches that can appear anywhere on the body, including the lower back and arms. The fact that the patches appear after scratching the regions suggests a possible allergic or irritant reaction, which is common in atopic dermatitis. Tiredness and migraine flares can be associated with the stress and discomfort of the condition.

Psoriasis: An autoimmune condition that causes scaly, itchy patches, often on the elbows, knees, scalp, and lower back. The sudden appearance and the fact that patches appear after scratching could be indicative of psoriasis, especially if there's a family history.
Contact Dermatitis: This is a type of inflammation of the skin that occurs when the skin comes into contact with an irritant or allergen, leading to itchy, scaly patches. Given that the patches appear after scratching, it's possible that the scratching itself introduces an allergen or irritant, or that the skin is reacting to something it has come into contact with.
Seborrheic Dermatitis: Characterized by a red, itchy rash, often on the scalp, but can also appear on oily areas of the body like the face, ears, and chest. However, it can sometimes appear on the arms and back, and the description could fit, especially if there's a history of similar rashes.

Cutaneous T-Cell Lymphoma (Mycosis Fungoides): A rare type of skin cancer that can cause patches of skin to become red, scaly, and itchy. Although less likely, missing this diagnosis could be fatal, so it's crucial to consider it, especially if the patches do not respond to typical treatments for eczema or psoriasis.
Infectious Diseases (e.g., Ringworm, Scabies): Certain infectious diseases can cause itchy, scaly patches. Scabies, for example, is highly contagious and can cause severe itching, especially at night. Ringworm (tinea corporis) can also cause itchy, circular patches. Both conditions require specific treatments and can spread if not addressed.

Pityriasis Rubra Pilaris: A rare skin disorder that causes constant itching and scaling (exfoliation) of the skin. It can start with small, itchy patches and progress to widespread scaling.
Keratosis Pilaris: A condition that causes small, rough, sandpapery patches on the skin, often on the arms, thighs, cheeks, and buttocks. It can sometimes be itchy, but it's generally less severe than the conditions described above.