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If it was a genuine marriage then it would help but the important thing, as I say, is that you have children who are UK citizens. However, it certainly cannot hurt and, actually, if you have proof that you have been together for 4 years then it will probably not make too much difference now because they will still be able to to see that you are in a relatinoship akin to marriage by virtue of the length of it and that you live together.
The registry office cannot refuse to marry you provided you can prove your identity and proof of address. They are not here to enforce immigration policy, indeed - they no longer require sight of a certificate of the approval for marriage which they used to.