Matthew Tvrdon, 31, of no fixed abode, admitted manslaughter on grounds of diminished responsibility. He also admitted seven attempted murder charges and one of dangerous driving.
Prosecutors will consult the victims' families over the next seven days.
At the moment Mr Tvrdon is still due to be tried next month.
Mr Tvrdon faces a total of 32 charges including counts of actual bodily harm with intent and grievous bodily harm with intent.
During the Cardiff Crown Court hearing, he also pleaded not guilty to murder and six counts of attempted murder.
His barrister told the court his client was suffering from paranoid schizophrenia.
Ms Menzies, 31, died on 19 October 2012 after Tvrdon ran her over at a fire station forecourt in Ely.
Others were injured in the incident which happened during the afternoon school run.
She was walking with two of her children when she was killed.
A van driver has admitted killing mother-of-three Karina Menzies in a series of hit-and-run incidents around Cardiff in October.