It's technically possible, but it would require a very specific context for C to be the natural, correct answer. The problem is that it does not follow the sequence of tenses. "who were living" is past progressive, so the following verb phrase, "may not appreciate," would also have to occur in the past. Children who were living in big cities *may not have appreciated...* would be possible. The meaning of answer C without changing the second verb phrase implies that the children are no longer living in cities, and they still do not appreciate animals. But the reason they do not appreciate animals' company is because they currently live in big cities where few animals are present.

Children who live/living in big cities may not appreciate the company of animals. It applies to the present time. Answer C however refers to the past.

Thank you.

When you say Children 'who were living' you mean those children are no longer living there now. For example, when you say 'she was a teacher', you mean she was a teacher in the past but she is doing something else now. Secondly, the word 'may' in the sentence suggests you to use present-related tenses. Therefore A and B are possible answers whereas C is not.