The correct option is A The cat's illness was making its paws sore.
Apostrophe is used to indicate possession. Apostrophe is used before 's' with singular nouns and after 's' with plural nouns. Here, 'cat' is a singular noun and illness belongs to the 'cat'. Also, our verb here is 'was' and that's a hint for us to identify the singular nature of our noun. Therefore, the correct option is: A) The cat's illness was making its paws sore.
We do not use an apostrophe in 'its' when we are using it to show possession. Here, we are using 'its' to refer to possession, that is, the cat's paws. So, we will not use an apostrophe for its.