Funding from the British Government’s anti-radicalisation programme has filtered through to extremist organisations that pose a threat to UK security, admitted Theresa May, the British home secretary.

May said the “flawed” programme she had inherited from Labour had financed “the very extremist organisations that [it] should have been confronting’’.

Speaking at the launch of a strategy review on countering extremism, she announced plans for future recipients of funding to be tested on their adherence to mainstream British values and subjected to stricter value-for-money controls.

The strategy has been delayed for months because of cabinet wranglings over the strength of the reforms. It updates a programme launched in 2007 to halt the spread of terrorism throughout the UK.

From now on the scheme, which has been allocated about £36m for 2011/12, will no longer provide funding to groups supporting “an extremist ideology” or expressing “extremist views”.

The review points out that “insufficient attention” has been paid to whether organisations receiving state funds subscribe to the rule of law, equality of opportunity and freedom of speech.

Aimed at remedying past mistakes, the new plan will also ensure that counter-extremism projects overseas, which are funded under the Prevent strategy but overseen by the Foreign and Commonwealth Office, are directly aligned with domestic priorities.

The document emphasises that while al-Qaeda is not the only terrorist organisation posing a risk to the UK, the organisation and its associates are the greatest threat.

Critical of the last government’s confused policies promoting integration with terrorism prevention bodies, May said that in future, integration policies would be kept separate from a revised Prevent strategy that promised to be “more focused, more rigorous and consequently more effective”

in dealing with extremism – including non-violent extremism.

The Street project, which works on de-radicalising young Muslims in association with a south London mosque, had its funding withdrawn this year on the basis of Ms May’s changes.

Yvette Cooper, shadow home secretary, accused May of “point-scoring” in her criticisms of the previous strategy. “If [Ms May] believes that she now knows all the answers on how to tackle extremism and radicalisation, she is heading for a fall,” Ms Cooper said.

Paul Thomas, an academic researching cohesion and anti-extremism at the University of Huddersfield, expressed further reservations about the Home Office proposals.

“The government is very clear that Prevent is not a way of spying on people, but the idea of [university teachers and health professionals] working on the programme sounds like a ramping up of the surveillance strategy,” he said. “This has got to be done by people who know what they are doing – how well trained are they?”

Meanwhile, Fiyaz Mughal, director of Faith Matters, a Prevent-funded organisation working on anti-extremism projects with prison chaplains, said the delay in publishing the review meant government-funded projects had come under insufficient scrutiny.