1. How long have you lawfully been in the UK? And on what visa? What country's passport do you hold? Were you married to the father of your children and if so, for how long?
2. I regret you will have to wait until you are five years on your visa, when you can apply for UK citizenship which would immediately make you free to get benefits. Until then, I regret to say that it will be impossible to collect social welfare benefits. Essentially, the visa has been issued on the basis that you will be able to support yourself or with the assistance of your UK citizen partner. I would advise you to seek maintenance payments from your former partner as he has a liability to pay maintenance to your for each of the child - 26% of his net after tax income. However, I regret to say that your visa terms would be violated if you sought to obtain benefits.
3. The UK Visa office does not grant variations allowing people to obtain benefits without revoking their visa. I am afraid this is an iron rule which is never broken. So you will have to wait until you get citizenship before you will be able to get benefits.
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