@Minae
if i remember correctly--you are filipina --right ?
i have some experience with immigration rules in the UK--when dealing with applications for settlement in the UK from the Philippines.
in the case of dependent children--it is more difficult for a child to apply for settlement in the UK with its mother--if the childs father IS named on the birth certificate. British immigration take the view that the father is jointly responsible for the child--unless the mother can prove otherwise.
i know this because my wife is filipina--and her boy has been refused a settlement visa in the UK because his father is alive and on his birth certificate.
in any case--i think you would need the fathers permission if you did want him on the birth certificate.