To find the angle between the minute hand and the hour hand at 4:25, follow these steps:

1. Minute Hand Position: The minute hand moves $360^\circ$ in 60 minutes, so it moves $6^\circ$ per minute. At 25 minutes, the minute hand has moved: $25 \times 6^\circ = 150^\circ$ from the 12 o'clock position.

2. Hour Hand Position: The hour hand moves $360^\circ$ in 12 hours, so it moves $30^\circ$ per hour. At 4:00, the hour hand is at $4 \times 30^\circ = 120^\circ$. In the next 25 minutes, the hour hand moves further. It moves $0.5^\circ$ per minute (since $30^\circ$ per hour $\div$ 60 minutes). In 25 minutes, the hour hand moves: $25 \times 0.5^\circ = 12.5^\circ$. So, at 4:25, the hour hand is at: $120^\circ + 12.5^\circ = 132.5^\circ$.

3. Angle Between the Hands: The difference between the minute hand and hour hand is: $150^\circ - 132.5^\circ = 17.5^\circ$.

Thus, the angle between the minute hand and hour hand is $17.5^\circ$.

Answer: C. $17.5^\circ$