The celebrity, perhaps best known for being half of 1980s pop duo Wham!, was arrested in July after he crashed his Land Rover into a photo processing shop in Hamstead, north London.
He has been banned from driving for five years and heavily fined after pleading guilty to the offences.
Initially, the singer was sent to Pentonville Prison to serve an eight-week sentence. However, he was transferred to Highpoint Prison in Suffolk after several days and then released after only being in custody for half the term of the sentence.
He is expected to serve the remainder of his sentence under the supervision of the probation service.
The 47-year-old has a previous conviction for a similar offence. He explained to reporters outside his home that he will "stop running away from you guys."
"I'm coming out here on my own so that you'll realise I just want to start again," he told reporters on Monday.
"By the way, I've just thought of a really good idea for a song," he added. "It's going to be nothing to do with prison."
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