The Scottish government is to "refresh" guidance about its named persons scheme after acknowledging "concerns".
MSPs debated the system, which will assign a named person to everyone under 18 in Scotland from 31 August, in a Conservative-led session at Holyrood.
The Tories sought consensus by calling for a "pause" in the policy, despite backing scrapping it altogether.
This was rejected by MSPs, who amended the motion to one backing the policy but agreeing "more must be done".
Holyrood voted to approve the named persons system by 103 votes to nil in 2014, as part of the Children and Young People Act.
It will see figures like family health visitors and senior teachers take on the "named person" role, which the government says will ensure children can get timely access to advice and services when they need it.
However, the Scottish Conservatives question whether the policy is "deliverable in the proposed format", and whether it is "in the best interests of children and families".
Education spokeswoman Liz Smith said the Tories had "deliberately steered clear" of explicitly opposing the policy as a whole in the debate, although the party still wants to see it scrapped.
She said: "The Scottish Conservatives believe there is growing parliamentary consensus for a major rethink.
"But there is also growing concern expressed by many frontline professionals - many of whom have no party political affiliation.
"We believe there is an urgent need to address the practical concerns of professionals and parents about the workability of the policy which, if it is not dealt with, could seriously undermine the welfare of children across Scotland."
Labour has also called for a "pause" in the roll-out of the named persons system, which has been trialled in some areas including Fife and the Highlands, to reassess public support for it.
The Lib Dems offered "cautious support" for the policy, while the Scottish Greens spoke strongly in favour of it - and against any pause - in the Holyrood chamber.
Deputy First Minister John Swinney offered some concession to opponents of the scheme, putting forward an amendment for debate underlining the government's support for the policy but acknowledging "the concerns that some people have expressed about the implementation".
The amendment, which was passed by MSPs to replace the wording of the Tory motion entirely, noted that "more must be done to ensure that implementation is successful, and that the Scottish government should refresh the guidance provided to professionals and the communication of the policy to the public".
Mr Swinney said the policy "embodies the principles we share as fundamental to supporting the lives of children and young people".
And he had strong words for the Conservatives, who he said had been "going around for months utterly misrepresenting" the policy.
When new Tory MSP Adam Tomkins warned of named persons working "behind the backs" of parents, Mr Swinney said he should be "ashamed", having "fuelled the absurdity of the attacks made on this policy".
Mid Fife and Glenrothes MSP Jenny Gilruth also accused the Conservatives of "using the death of a child" in her constituency - referring to murdered toddler Liam Fee - "to score political points".
Labour's Iain Gray, who also attacked the Tories for "opportunism", said his party backed the named persons system "if implemented properly and proportionally".
He said the government admitting some parents had concerns with the implementation was "a big, big step in the right direction".
However, he said the government had allowed "wild and wilful distortions of what this policy could mean to run unchecked", calling for a "pause" in implementation while an external review is carried out.
His amendment to this end was rejected by MSPs.
Lib Dem Tavish Scott also voiced some support for the policy, saying parents and carers could "say thanks but no thanks" to their child's named person.
However, he lodged an amendment to Mr Swinney's contribution noting the concerns of health professionals and school staff about resources needed for implementation, which was accepted after a vote.
The 'No to Named Persons' campaign group criticised the government's concession as "rearranging the deck chairs on the Titanic".
Spokesman Simon Calvert said: "What is being proposed is simply a waste of time. They are seeking to refresh their publicity campaign which has thus far proven to be a disastrous failure, clearly losing parental confidence."