How valid is the statement that an illegitimate child prevents his father from entering Paradise?

This statement is incorrect. If an adulterer repents with sincere repentance, his repentance will be accepted, and he will enter Paradise. The fact that he has a child born of adultery does not prevent him from entering Paradise. This is supported by numerous evidences from the Quran and Sunnah regarding the acceptance of the repentance of those who commit major sins and their entry into Paradise. This is the doctrine of the Ahl al-Sunnah wa al-Jama'ah (Sunni Muslims). Al-Bayhaqi narrated from Ibn Abbas that the verses from Surah Al-Furqan (verses 68-70) and Surah Az-Zumar (verse 53) were revealed concerning people from the pre-Islamic era (Jahiliyyah) who had committed these major sins and then repented. Nevertheless, it must be known that adultery is a major sin, and its perpetrator deserves a severe warning and entry into Hellfire if he does not repent or if Allah does not forgive him.