Revealing mistake: When Supergirl falls on the planet ground and gets released from the mirror trap, she gets up and the hard floor stretches down, revealing it's a rubber mat to soften her fall.
Revealing mistake: When Selena uses her magical powers to accelerate the fairground ride to superfast speed, the twinkling fairground lights in the background are also flashing much faster. They obviously played the tape back at high speed to achieve this.
Revealing mistake: When Supergirl is flying to Earth, part of her head disappears due to the blue screen used. Also, some sides of the spaceship either disappear or become transparent.
Answer: The Omegaahedron is the power source for the whole city. It supplies the air, the dome that protects them, the lights, and the machines they need to survive.
The question was why would Zoltar need permission to use the Omegahedron, not what it could do.