He was named after biscuit companies, due to Jonny's love of biscuits (and according to Jonny he smelled of biscuits as stated at Corinthian's birth.) It was found that Gaz Wilkinson was his real father in the episode Here Bums the Bride. When Gaz donated sperm, he was found to be of low sperm count, making Corinthian unlikely to be his. However, Jonny took a DNA test (off-screen) before he died, and found that Corinthian was not his.
Janet still thinks that Corinthian is Jonny's, and refuses to believe otherwise. This is most noteable in "The Aftermath: Part 1", where she said, she'd "sacrifice Gaz" rather than lose Jonny (by finding Corinthian wasn't his). She has not yet seen the DNA test.