The doctor gave me a prescription . . . . . . . . some medicine yesterday.

Option A, 'With', refers to 'in the company of; alongside; close to; near to'.
Therefore, option A is incorrect as the given sentence does not imply being near or alongside the cough of the subject.
Option B, 'To', refers to 'indicating destination: In the direction of, and arriving at'.
Therefore, option B is incorrect as the given sentence does not indicate any destination or arrival related to the cough.
Option C, 'For', refers to 'because of'. Therefore, option C is correct as it indicates that the given prescription of the subject in the sentence was made because of the subject's disease which is unknown. Option D, 'At', refers to 'in or very near a particular place'.
Therefore, option D is incorrect as the given sentence does not indicate the prescription being near or in a place.