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Probably the commonest condition to present with raised itchy white bumps is Molluscum Contagiosum
. The second possibility would be Keratosis Pilaris but here more than white bumps the lesions are like small rough spikes in the hair follicles.
Molluscum Contagiosum is an infectious skin condition caused by a virus (Pox Virus) and spreads by skin to skin contact. Mollusca are self resolving but can take 6 months to 2 years to disappear.
The itching in Mollusca is because of an allergic reaction to the virus and applying a 1:1 mixture of 1% Cortizone cream with Bacitracin ointment on the bumps twice a day will help as will taking Tab Claritin 10-20 mg per day depending on the severity of the itching. All of these are available over the counter without a prescription.
I would suggest showing the bumps to your doctor in person and discussing the possibility of Molluscum Contagiosum. If confirmed you could either leave them alone for spontaneous resolution or have them removed by either radiofrequency, laser ablation or needle extirpation followed by TCA application.
NOTE: The virus can spread by auto-inoculation. That is by scratching existing lesions with the fingers. So if you're not having them removed, do not touch the mollusca unnecessarily.
Hope this information is useful, please feel free to ask if you have any queries.