All three young children were said to have suffered from spinal muscular atrophy. Also known as floppy baby syndrome, the genetic condition leaves children with little control of their movements and can drastically shorten life expectancy.
Mr Clarence, who works at the City bank Investec, is said to have been away in the family's native South Africa with their eldest daughter, who does not have any health problems, at the time of the deaths.
Chairwoman of the bench Fiona Abbott said: "For these three charges you will be sent to the Central Criminal Court at the Old Bailey for a preliminary hearing on 9 May.
"There will be a bail application at the Central Criminal Court at 10am on 29 April."
Mrs Clarence is originally from South Africa and moved to Britain with her husband some years ago.
Additional reporting from PA