Alan Creeton, 41, had secretly been paying pounds 20 a week maintenance to the woman.
But, when a CSA assessment was posted to him, he realised his wife Christine would find out, the deputy Southampton coroner Christopher Hodgkinson was told.
Mrs Creeton had no idea about the 12-month-old child - but often questioned her husband as to why they could never afford a holiday.
The coroner heard how, after the CSA letter arrived on July 25, Mr Creeton visited Louise Powell, a trainee legal executive, to ask if the matter could be kept from his wife - and mentioned suicide.
The following day he saw his wife at Ringwood Sports Centre, Hampshire, where she worked.
He then left a letter at their home in Christchurch, Dorset, saying that the last two years had been a nightmare and that, by the time his wife read the note, he would be dead.
Mr Creeton drove to a remote spot in the New Forest and killed himself in his car by running a hosepipe from a window to the exhaust.
Miss Powell told the coroner: 'He said he had been named by the Child Support Agency as the father of a child and asked how much he would have to pay.
'He asked if could he keep it from his wife. I told him that the CSA would send all the details to his home.'
'As he left my office he said maybe he should commit suicide.'
Mr Hodgkinson, recording a verdict of suicide, said: 'Mr Creeton was worried about maintenance payments and his wife finding out about them. He had also received paperwork from the CSA that was worrying him.'
After today's hearing Mrs Creeton refused to discuss her husband's relationship with the other woman or the secret child.
'My husband has died - and he died to keep a secret,' she said.