angel786 wrote:as people on here are confused with differences of british citizenship and british passport
One needs to be a British citizen before one can apply for a British passport.
As per OP the father is not listed in the child's birth certificate. So the question is, whether OP has any other acceptable documentary evidence to prove that the British citizen (father) is actually the father of the child? If there is none, the relevant link - explaining how the father's name can be added to the birth record of the child - has also been provided.
Since the OP (mother of the child) is not a British citizen, the only way the child can acquire British citizenship is through the British citizen father. If the OP has documentary evidence of the relationship of the child with the British citizen father (through whom the child may have automatic acquisition of British citizenship), only then an application for British passport can be submitted.