Answer: B — $3$

Let the number of students in the class be $n$.

The first correction changes a student's score from $0$ to $100$, so the total score increases by $100$. Since the average increases by $4$, we have:

$\frac{100}{n}=4$

Therefore:

$n=25$

Now, another student's score was recorded as $81$ instead of $56$. Correcting this decreases the total score by:

$81-56=25$

Since there are $25$ students, the average decreases by:

$\frac{25}{25}=1$

After the first correction, the average becomes $x+4$. After the second correction, the final corrected average is:

$y=x+4-1=x+3$

Hence:

$y-x=3$

Therefore, the correct answer is $3$.