@Thorgen is right. Because you´ve already been granted asylum in Italy, your renewed asylum application in Germany will probably rejected as inadmissible.
As a father of an underage child who does not have the German citizenship, you can try to apply for a residence permit, but you´re not entitled to get it granted. The decision on your application is at the discretion of the immigration office.The residence permit is usually granted only if the family unity between you, your child and his mother can only be conducted in Germany. It must be unreasonable for your child and his mother to follow you to your homeland or to Italy. According to the Federal Administrative Court, the fact that a family member, in this case, your child and his mother, has a residence permit, doesn´t preclude a joint departure.
If you are in possession of an Italian permit for permanent residence - EU ("soggiornante di lungo periodo -- CE od. UE"), then you can enter Germany without a visa and apply for the residence permit at the immigration office within 90 days after your entry according to § 38a of the Residence Act. A prerequisite for this is, for example, the submission of a concrete fixed job offer with which you can earn sufficient income to secure your livelihood. In this case, the granting of a residence permit in accordance with § 38a of the Residence Act requires that there are no suitable German job seekers and foreign job seekers entitled to reside for the work you intend to do and that your payment is comparable with that for German employees. Information on this topic can be found at the following link of the Berlin immigration office: