For me, it sounds like "a height" with article "a." Is the script right? And can I use "a" in front of "height" in general?
Yes, if you listen carefully, he does say "add" and not "a" here. The script is correct. You can use "a height" depending on the context, but it cannot be used generally as "a height" itself is not grammatically correct. For example, "a height" is acceptable when it is used in combination with "from a height," which is a general expression used to mean "from a high position/above." Ex: The helicopter can show tourists the skyline from a height. Ex: The architect the building at a height that had not been attempted before.