those reasons IMHO, are VALID reasons for you to keep sole custody. I have a family member who did exactly what this man has done. he has at least 7 children, and supports none of them. The family only has contact with one of his children, and only because I initiated the contact when I was 20 and she was 12.
However, if he wants to be in your son's life, the court will allow it, You need to accept that and offer him some access. It will really help your case to show that you recognize his father and have made attempts to invite him into your son's life. If you prefer them to be unsupervised visits, you will need to either trust the people he chooses to supervise, or arrange for someone else. An offer even of supervised visits is better than no offer at all.
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