The revelation was made during a bail application by the 23-year-old man accused of killing the youngster.
Liam Whoriskey, from Glenabbey Gardens in the city, faced the court via videolink from Maghaberry prison on Monday.
District Judge Barney McElhome heard that the accused was said to have been drinking while babysitting three-year-old Kayden and the toddler’s baby sister in a flat at Colmcille Court in the Bogside last September.
He was told that Whoriskey informed told police during interviews at Strand Road PSNI station that he had put Kayden to bed and heard him laughing and playing – but when he checked on him the next morning, the little boy was dead.
Asked how Kayden had sustained two black eyes in pictures circulated on social media, Whoriskey suggested the child had hit himself with his own toys.
Whoriskey, who has previous convictions for domestic violence, denies murdering the toddler.
The court also heard that tensions were running high in the Colmcille Court and that Whoriskey was now the subject of a paramilitary threat.
District Judge McElhome refused to release Whoriskey into the community because of the tensions and the threat to his life.
Whoriskey was remanded in custody to reappear in court again next month for a further remand hearing.