The court was told that Sheppard was driving home after collecting her sons from school when the crash happened on a bend.
Prosecutor David Eager said the “the seeds were sown the previous day” when Sheppard sent her partner a message that said “my tyre is going flat again”.
Mr Eager said Sheppard’s partner mistakenly checked her front right tyre that evening.
“Sadly, tragically, he did not check any of the other tyres,” said Mr Eager.
CCTV footage showed the under-inflated tyre on Sheppard’s journey to work and Mr Eager explained there were two local filling stations with with air machines within a four-mile (6km) radius when she started her last journey.
The court heard a driver following the car before it crashed spotted the under inflated tyre and said Sheppard’s vehicle was “wallowing.”
“The back end went, it turned 45 degrees,” the driver said.
One of her sons, Louie died at the scene and her other son, Mason, along with the other child, were airlifted to Sheffield Children’s Hospital, where Mason died a few hours later.
Mr Eager said Sheppard, who was also injured, admitted in a police interview of being aware of the under-inflated tyre and not rectifying it.