There is a reason why you don't see homeless families starving on the streets in the UK. There is a social safety net, even for overstayers. Seek that help. There is no reason for your family to end up homeless in the UK.
It's disconcerting the direction this thread has taken. Don't lie. Don't ever lie. Not only might you destroy the life of your partner for no good reason, if you get caught lying, you can probably forget about ever getting to stay legally in the UK. That said, do consider the situation you are in and how you got there. You've got atleast two very good reasons not to protect anybody.
A few basic questions:
- * Are you married, or were you ever married, to the father of the children?
* Is he a citizen, or is he a permanent resident (which is it)?
* Is he currently in the UK?
* What are the ages of the children?
* How long have you been in the UK?
* On what basis did you originally come to the UK (was it because of your partner)?