This is Halfway Across the Galaxy and Turn Left (1994–1995).
It's an Australian TV show based on a 1985 novel of the same name by Robin Klein. She wrote one sequel, titled Turn Right For Zyrgon (1994).
According to the Wikipedia page, the lead character is a 12-year-old girl named 'X' who has an older sister, a younger brother, and a mother and father. They get in trouble with the authorities on their home planet of Zyrgon when the father cheats the government lottery for the 27th time in a row, and end up seeking refuge on Earth:
The novel tells the tale of the planet Zyrgon, ruled by the galactic police called The Law-Enforcers. They are after Mortimer, who has cheated the government lottery for the 27th time in a row. His family is governed by the youngest daughter, 12-year-old X, who wants to save her father from the detention centre.
The family also includes Mother, who would rather design clothing and leave all worries to her daughter X. The oldest sister Dovis is a cosmic flier who writes poetry and levitates. The youngest is a boy genius, Qwrk who is a professor at age 8.
X is the lead character: a stressed girl who has to balance between strange Earth customs such as school and her duty to take care of her family.
Zyrgonians have special powers such as levitation, simulations, and kinetics. They love gambling and live on an ultra-modern and dystopian planetoid.
I found quite a detailed episode guide here. The plot summary for the first episode mentions that X was awarded the scholarship for Excellence in Family Organisation:
On the far away, frozen planet of Zyrgon, twelve-year-old X is awarded the scholarship for Excellence in Family Organisation but her excitement is short-lived after her father manipulates the government lottery and wins for the 27th time in a row. With the law threatening the family with jail — or worse — X uses her status as the new Family Organiser to lead her family halfway across the galaxy to their new home — on Earth.
The plot summary for the eleventh episode mentions that X's father manipulated her test scores and that the family were condemned to lava:
Explaining that she sympathises with X as a fellow organiser, Jady releases Hecla and Lox from custody and tells them the family have arrived home. Hecla and Lox go to the family's home but the Chief is waiting for them and condemns them to the lava. Rescued by friends, Lox contacts X via thought-beam to tell her he and Hecla are safe. When he tells X she didn't win the scholarship fairly, father admits to manipulating the scores. With Hecla safe, Father suggests to X they stay to Earth and X agrees.