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Questions have been raised over what action the first minister took after Health Minister Eluned Morgan was banned from driving for six months.
Mark Drakeford has said he has dealt with the issue under the ministerial code and the matter was "closed".
In a letter the Welsh Conservatives demanded details of how Mr Drakeford came to that conclusion, and how he found out about the offences.
The Welsh government said it would respond to the letter in due course.
Mold Magistrates Court banned Ms Morgan, Labour Member of the Senedd (MS) for Mid and West Wales, for speeding at a hearing in March.
A report by the Senedd's watchdog, standards commissioner Douglas Bain, said it had followed three other speeding convictions and resulted in a £800 fine.
Mr Bain's report, which has not been published but has been seen by BBC Wales, found Ms Morgan had breached the Senedd's code of conduct - rules that govern their behaviour - and had showed a disregard for the law.
His findings mean she could be subject to a further punishment such as a temporary ban from the Welsh parliament, to be considered by the Senedd's standards committee.
Politicians in the Welsh government are also subject to a separate set of rules on ministers' behaviour - called the ministerial code.
It is up to the first minister to decide how complaints under the code are investigated. The code says he will "usually refer significant complaints regarding ministerial conduct to an independent adviser for consideration and advice".
On Wednesday Mr Drakeford said: "I have dealt with it under the ministerial code and the matter is closed."
The Welsh Conservatives said there were "a lot of questions to be answered," adding: "Labour are clear that politicians who break the law should step down - unless of course they're Labour ministers in Wales."
In his letter to the first minister Welsh Tory Senedd leader Andrew RT Davies asked whether Ms Morgan referred herself for investigation under the ministerial code, or whether Mr Drakeford initiated the investigation, and when Mr Drakeford became aware of each speeding fine.
He asked when his investigation began, and what evidence did he consider when investigating the fines.
He added if he sought independent legal advice, and "on what basis did you determine that your minister had not broken the ministerial code".
🥀Labour are clear that politicians who break the law should step down - unless of course they're Labour Ministers in Wales.
❓ There are a lot of questions to be answered about the health minister's recent driving ban.
👇 Read @AndrewRTDavies' letter to the First Minster. pic.twitter.com/0yBfvwcZCQ
The BBC is not responsible for the content of external sites.View original tweet on Twitter
Welsh government ministers have been highly critical of the UK government over partygate.
Mr Drakeford called for Prime Minister Boris Johnson to resign after he received a fixed penalty notice for breaching Covid rules, saying: "You can't be a law maker and a law-breaker at the same time."
On Wednesday, Mr Drakeford said it was right the matter was taken seriously, and said: "She didn't make the law that she has broken. She admitted it, immediately. She has been dealt with by the courts.
"There is no moral equivalence [with] the prime minister in making laws himself that he went on to break, denying that he broke them, provoked a long and expensive police investigation to reveal the fact that he had himself had indeed broke the laws that he had made."
Mr Drakeford said he had not seen what the standards commissioner had said.