Thanks for your question. I will try to help.
A case worker will not normally refuse an application where the only good character issue is non-payment of council tax because of your financial position unless you have unreasonably refused to pay or have made some false statement to avoid paying the correct council tax liability.
So, assuming these do not apply to you then you should be okay.
A liability order is effectively a legal demand for payment, rather than a judgement made by court after contentious proceedings. However, I would probably disclose the liability order but mitigate it by making a statement confirming what has happened and showing evidence of the payment plan and you having kept to the payment plan.
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