From what you say, you have been living in a relationship akin to marriage with your current partner for eight years and have regular contact with your child who is settled in the United Kingdom. You can therefore make an application from within the United Kingdom to remain with your current partner without the need for having to leave the United Kingdom despite your current immigration status. Your application will also invoke your rights under article 8 of the Human Rights Act 1998 (as incorporated) and so long as you can satisfactorily evidence what you claim to be are the facts of your case and that you can support and maintain yourself in the United Kingdom without recourse to public funds, then you have every chance of successfully obtaining stay in the United Kingdom
Ideally you should instruct a good immigration solicitor to help you through the process.
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