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I assume that you are receiving qualifying benefit such as DLA MRC or HRC, Attendance Allowance or a PIP Daily Living award.
To receive SDP you need to be living alone or considered to be living alone, I think you will struggle to show that the Care Worker is not living with you as I assume that they are there to provide you with care.
However, you have nothing to lose by trying. You will need to show that they are living separately from you, typically this means their own rent book and that they have their own rooms in the property and only a kitchen and bathroom are shared.
Gordon