Can the father of my child file with her this year

File for a SSI number for the child if you haven't already... and yes, with the Number you can claim them as a dependent when you file the year of their birth.

I am assuming that the parents do not live together with the child (please correct me if I am wrong). The father can claim the child as his dependent only if he is the custodial parent. Otherwise, if you are the custodial parent you would claim the child as your dependent and the non-custodial parent (the father) can't. However, if you do not have reason to file a tax return, then you can in effect give the father permission to claim the child as his dependent.