Ask an Immigration Solicitor. Get an Answer ASAP.
Children born in the UK while the parents had valid leave to remain do not need to do anything ONLY if they have not left the UK and have not applied to regularise their status. If you left the UK with your children and return with them, I am afraid the advice you received not to include them in your visa was wrong.
You must inform the Home Office where your children are
How do you know the Home Office wants to know your children's whereabouts.
OK reply to the letter explaining where your children are
You asked whether to send a letter and i answered your question
This is your question. Please be honest.
"How do I do that? Send a form if yes what form, a letter or what other means to inform the home office and what documentation is required? Thanks"