This is likely I Shall Master This Family, aka I'll Be the Matriarch in This Life. It is licensed by Tapas.
The great Lombardi family was once at the top of the empire. After the death of its patriarch, the fate of the family and that of Firentia, born from a Lombardi and a peasant, fall to ruin at the hands of her useless and cruel cousins. But when she’s reincarnated as her seven-year-old self, she’ll work to protect the family’s honor, gain her grandpa Lulac’s favor, and prevent her own father’s death. In this lifetime, there’s only one way for her to win: become the head of their mighty household.
It's not mentioned in the synopsis but she is also a transmigrator, and lived her first life in Korea. She reincarnates as the illegitimate daughter of a prominent wealthy family.
Most others in her family are extremely biased towards noble blood and treat her like dirt. In her second life she is kicked out of the family after the grandfather dies; her uncles and cousins promptly destroy the family's wealth and reputation within 2 years. She gets drunk and run over by a carriage, and reincarnates as her 7-year old self.
She then proceeds to become the matriarch of the Lombardi family in order to save it from her incompetent family members and also the lives of her father and grandfather.