The correct option is D has
The pronouns such as 'each', 'either' and 'neither' are of singular nature and take a singular verb. 'Either' means any one of the two things/people. These pronouns are usually used with two elements to choose from. Since, both elements, the child and the mother, are singular, the sentence will take a singular verb. Hence, Option D, 'has' is the correct answer. Option B and C are plural forms, so they are incorrect.