1. **State the problem:** The admission fee for a child $c$ is $4$ more than one-half the admission fee for an adult $a$. We want to find the total amount paid for 12 children and 4 adults. 2. **Write the expression for the child's fee:** Given $c = \frac{1}{2}a + 4$. 3. **Calculate total cost for children:** Total for 12 children is $12c = 12\left(\frac{1}{2}a + 4\right)$. 4. **Simplify the children's total:** $$12\left(\frac{1}{2}a + 4\right) = 12 \times \frac{1}{2}a + 12 \times 4 = 6a + 48$$ 5. **Calculate total cost for adults:** Total for 4 adults is $4a$. 6. **Calculate total amount paid:** $$6a + 48 + 4a = (6a + 4a) + 48 = 10a + 48$$ 7. **Answer:** The expression describing the total amount paid is $\boxed{10a + 48}$, which corresponds to option C.