I can understand David Cameron’s frustration this week. A win at Newark (albeit in the 40th safest Tory seat) and the Queen’s speech (even a thin, feeble one) have been overshadowed by the egos and leadership ambitions of two of his cabinet.

The Gove-May spat is pretty breathtaking even for those of us who have been in government when people did not always see eye to eye. Neither comes out well. Briefing your former workmates at the Times against a fellow minister and a senior official is bad government. Equally, while I have seen plenty of intra-cabinet letters written with the intent to leak if necessary, I have never seen one so brazenly trumpeted and even placed on the departmental website.

But while these shenanigans are interesting, even entertaining, the real damage done by this row is to the Prevent policy which needs to be a central part of an effective counter-terror strategy. As home secretary I published the first public counter-terror strategy. It built on the four ‘Ps’ developed by John Reid: Pursue, Prepare, Protect and Prevent. To deal with the terror threat, which is still very real, we need to be able to pursue and catch those plotting and executing terror attacks; we need to prepare for attacks; we need to protect our borders, our public spaces and other potential terror targets.

However, when much of the terror threat is fuelled by an extreme Islamist ideology, it is also necessary to prevent people becoming radicalised and potentially violent in the first place. This is the principle at the heart of the Prevent strategy which we worked hard on and, incidentally, argued passionately about when we were in government. I led the major development of the strategy with Hazel Blears. There was a lot of discussion across government about the extent to which we should challenge the extremism which created the ‘oxygen’ for violence and radicalisation even before it became explicitly violent; there was concern about the partners we worked with; there were arguments about how the money should be allocated and accounted for. On the whole, those arguments were carried out within government and in a comradely manner.

The result of the work was a comprehensive and well-funded programme. It had breadth, recognising the need to work in communities, in mosques, in schools and universities, in prisons, with international partners and for the police to engage in neighbourhoods as well as in the specialist counter-terror units. It supported mainstream Muslim voices, community groups, local partners and initiatives; it recognised the need to identify the spaces where radicalisation could occur; where necessary, it allowed grievances to be debated and addressed.

But it was also at the most difficult and controversial end of public policy. There were some risks taken and some difficult judgements made. Some important partners remained sceptical. It was a work in progress, but instead of building on it and improving where necessary, the new coalition government trashed the existing programme and cut its funding.

This was a far greater setback than any bickering between ministers. The Trojan Horse issue playing out in Birmingham schools demonstrates just how difficult, sensitive and current the issue is. The response has been piecemeal and partisan when it needs to be cross-party and inclusive. This is a lesson we will need to reflect on carefully if we get the opportunity to pick up the pieces of this crucial policy next May.

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Jacqui Smith is a former home secretary, writes the Monday Politics column for Progress, and tweets @smithjj62