When exactly do the lights get turned on in the second play? Do you press the space key again in that moment? Or does it suddenly get turned on (and off) for no apparent reason?
Have you already compared your code to the Lecture Project Changes which can be found in the Resources of this lecture? If so, please share your code here.
And which version of Unity do you use? There is always a chance that there is a bug in Unity. Updating to a newer stable version might help.
See also:
- Forum User Guides : How to apply code formatting within your post