The result comes after the new First Minister finalised his cabinet and wider ministerial team by including a minister for independence, a move that has angered pro-union campaigners and led to a complaint to the top civil servant by the Scottish Conservatives.
In what is a painful judgement on his time in government, the public also suggest that his time spent as a minister was, on the whole, a failure and that they have little faith in the new SNP leader to improve public services during his time in the top job.
Scots are also in favour, by plurality, of there being a snap Scottish Parliamentary election following the change of First Minister, with 49 per cent of Scots wanting to see one, and 36 per cent opposing one, with 15 per cent stating they did not know.
The result, which will be a boost to Scottish Labour leader Anas Sarwar, who used the resignation of Nicola Sturgeon and appointment of Mr Yousaf as an excuse to demand an election, also saw 45 per cent of SNP voters agree, with 40 per cent stating they were opposed to a snap election.
The poll, undertaken by Savanta for The Scotsman, interviewed online 1,009 Scottish adults aged 16 or over between Tuesday, March 28, and Friday, March 31.
On Friday, Mr Yousaf said he would be a First Minister to govern for “all of Scotland”, adding: “your priorities will be my priorities.
"I will work tirelessly to serve you, the people of Scotland. I hope I can do you proud.”
The poll asked the public whether Mr Yousaf was too focused on domestic policies or on achieving Scottish independence, with 30 per cent of all voters stating he was “far too focused” on independence and “not focused at all” on domestic priorities. A further 15 per cent said he was “slightly too focused” on independence, and “not focused enough” of domestic issues.
A quarter of voters, 26 per cent, said the SNP leader had “appropriately focused on both” sets of priorities, while five per cent said he was not focused enough on achieving independence, and four per cent said he was not focused at all on that priority.
A further 21 per cent, around a fifth, said they did not know.
Overall, 45 per cent of voters said Mr Yousaf was too focused on independence, with just eight per cent (due to rounding) stating he was too focused on domestic issues.
Around half, 44 per cent, of SNP voters think their party’s new leader is “appropriately focused” but more think he is too focused on independence than not focused enough, 23 per cent versus 12 per cent.
The new First Minister’s record in government also took a beating in the poll, with 41 per cent of Scots stating his record as justice and health secretaries was a failure, including a quarter (25 per cent) of SNP voters.
Just 22 per cent of Scots said his ministerial career had been a success, including a third of SNP voters (34 per cent).
A quarter of voters (23 per cent) said it was neither a failure or a success, and 14 per cent said they did not know.
He is also not fully trusted to make public services better, with over a third of Scots (36 per cent) believing they will get worse while he is First Minister, and a further third (31 per cent) believing they will stay the same.
Just 20 per cent of Scots believe public services will improve under Mr Yousaf, including just over a quarter of SNP voters (27 per cent).
Mr Yousaf’s election as First Minister also received mixed backing, albeit driven by party sentiment, with 43 per cent stating he was the wrong choice and 32 per cent stating he was the right choice. A quarter said they did not know.
His position on challenging the Section 35 order which has blocked the passage of the Scottish Parliament’s Gender Recognition Reform bill is also not universally backed, with 45 per cent stating he should not to do so. Just 16 per cent said he should do so no matter what advice he receives from the law officers, while 17 per cent said he should do so provided he is not told it is unlikely to succeed.
A fifth, 21 per cent, said they did not know.
In a boost to both Mr Yousaf and his coalition partners the Scottish Greens, however, the Scottish public is mostly supportive of the smaller party’s return to government.
A third of Scots (35 per cent) said the Greens remaining in government was a bad thing for Scotland, compared to 44 per cent who said it was a good thing for Scotland, including 66 per cent of SNP voters. A further 20 per cent of Scots said they did not know.
Mr Yousaf’s approach to independence policy is also backed by the public, despite a third (35 per cent) stating the powers to hold an independence referendum should not be passed to Scotland under any circumstances.
A fifth, 21 per cent, believe the powers should be handed over permanently regardless of any election outcome, while 12 per cent said a majority of seats at the next general election was enough for the powers to be transferred, while seven and six per cent said the SNP being largest party or winning a majority respectively at a Holyrood election would constitute a mandate to do so.
A fifth, 19 per cent, said they did not know.
Chris Hopkins, political research director at Savanta, said: “Off the back of a leadership campaign where it felt, at least to an English observer, that Scottish independence was not the primary issue, it’s interesting to see twice as many SNP voters say that Yousaf’s priorities are too heavily weighted towards independence and not enough towards domestic policy.
"His record in government, while viewed as a success by a third of SNP voters, is seen as a failure by a significant quarter of them, and therefore it does appear as if Yousaf still has plenty to do to convince his party’s supporters that he’s the right person for the job.
"Many will see it as important that Yousaf creates his own mandate, yet SNP supporters saying there should be an early Holyrood election may be careful what they wish for, as the only thing Yousaf could create is a rod for his own back, based on the latest polling numbers.”