Hello
1. it is your child, but this needs fast legal prove! Due that you are british contact the embassy and ask them on details on your fathership. In General it will require a genetic test, you need to know which lab has the registration by the british government, it will be most likly outside ethiopia. You will have to go and see a registrated doctor in Addis (most likly at the nordic clinic) who will take all of zour three bloed, the samples are send out to Britain etc...is quite expensive, can go up to some 400 Euro.
2. if you once the father, it is difficult to reject a residence for you, parents can not be seperated so easy from the children!
3. you should think to get the child as well british, will make again your stay easier
I went through all this, but I am German, so I can only refer to the German law, but I belive it should be not such a big difference!
4. Find a Job, for sure very difficult and depending a lot on your qualification.
Oliver