Finance Secretary Mark Drakeford has said the first minister has his full support “while he is willing to go on”, following the death of Carl Sargeant.
The Labour AM was found dead four days after being sacked from the cabinet over personal conduct allegations.
Mr Drakeford told BBC Radio Wales Carwyn Jones will have asked himself “time after time” if he could have done things differently.
He also said he did not “recognise” claims of bullying within government.
An inquest on Monday heard the provisional cause of Mr Sargeant’s death at his home in Connah’s Quay, Flintshire, last Tuesday was “hanging”, described as “an apparent act of self-harm.”
Mr Jones has announced an independent inquiry into events surrounding the tragedy, but has faced calls to resign over his handling of the matter.
Speaking on Good Evening Wales, Mr Drakeford said: “I’m quite sure the first minister will have asked himself time after time as to whether there’s anything he could or should have done differently.”
Referring to the inquiry, he added: “Now there will be someone who’s not been involved in it at all who will ask those same questions.
“And I think we’re better off now leaving it for that process to unfold.”
Asked if the first minister had his full confidence, Mr Drakeford said: “I think he has faced an incredibly difficult week personally.
“I think he is doing his very best to respond honestly to the concerns that are raised.
“I don’t think raising questions about his future in any way helps to resolve some of the matters that we have discussed.
“He has been a very successful first minister and, while he is willing to go on bearing the very considerable personal burdens that all this places on him, he will certainly have my full support.”
Earlier, former Labour AM Jeff Cuthbert – who served in the cabinet as communities minister from 2013 to 2014 – said he too did not recognise claims of a bullying culture within the administration.
Steve Jones, a former special adviser to Carwyn Jones, had said he agreed with ex-public affairs minister Leighton Andrews that there was a “toxic” atmosphere, and claimed some ministers were “undermined by senior advisers playing power games”.
However, Mr Cuthbert – now Gwent police and crime commissioner – said that was not an “accurate reflection” of the situation.
“I generally found it a friendly or – as we say in the Labour party – comradely atmosphere,” Mr Cuthbert told BBC Wales.
“As government ministers we’re all capable of being passionate, we’re all capable of being robust when we felt our intentions, our policies or initiatives were not being taken as seriously by others as we would have liked.
“So we would have our arguments from time to time, but that’s to be expected at government level.
“But quite frankly it never got personal or any way intimidating … that’s my recollection.”
The Welsh Government has also said it did not recognise the claims made by Steve Jones.