On a balcony that is partially covered by a roof, can the sukka be extended under the covered part?
Only the part under the sky (not under the roof) is valid. If the part outside the roof has three walls and valid schach, it is treated as if that is the succah and the part under the roof does not exist.
Another possibility is discussed in How far can a sukkah's walls be from schach? which speaks about the distance from the wall that invalid schach can exist without invalidating that wall.
Of course your local Orthodox rabbi should see the succah to explain the details to you.
As we see in Building A Sukkah - Advanced
- What if one part of the Sukkah is under the sky and one part is not?
If the section under the sky has sufficient walls and the minimum dimensions, the Sukkah is kosher. Nevertheless, one may only eat in the part of the Sukkah that is under the sky. Sometimes, the invalid section may be included in calculating the size of the Sukkah and a rabbi should be consulted