To solve this question, we need to determine how much the hour hand of the clock moves in 10 minutes past 5 o'clock.

1. Calculate the total degrees covered by the hour hand in 12 hours:
- In 12 hours, the hour hand completes a full circle, which is 360 degrees.
- So, in 1 hour, the hour hand covers 360 degrees / 12 = 30 degrees.

2. Calculate the degrees covered by the hour hand in 1 minute:
- In 1 hour, the hour hand covers 30 degrees.
- So, in 1 minute, the hour hand covers 30 degrees / 60 = 0.5 degrees.

3. Calculate the degrees covered by the hour hand in 5 o'clock:
- At 5 o'clock, the hour hand is pointing directly at the 5 on the clock.
- Since the hour hand moves from one number to the next as the minutes progress, it covers an additional 0.5 degrees for every minute.
- From 5 o'clock to 5:10, there are 10 minutes, so the hour hand covers an additional 0.5 degrees * 10 = 5 degrees.

4. Calculate the total degrees covered by the hour hand in 10 minutes past 5:
- At 5 o'clock, the hour hand is at 5 degrees.
- With an additional 5 degrees covered in 10 minutes, the hour hand has turned through a total of 5 degrees + 5 degrees = 10 degrees.

5. Convert the degrees covered by the hour hand into minutes:
- Since the hour hand moves at a rate of 0.5 degrees per minute, we can convert the 10 degrees covered into 10 degrees / 0.5 degrees per minute = 20 minutes.

Therefore, in 10 minutes past 5, the hour hand has turned through 20 minutes.