On last week's trip to Washington, David Cameron brought along the usual team of political correspondents, and one less likely reporter. She was from Grazia, the women's magazine, and was given lots of special access. Expect more spreads showing Sam Cam's "great" fashion sense in the next issue. It may be a trivial thing, but it reminds us just how worried Cameron is about the female vote.

Last year the Tories were falling behind when it came to women voters and there are signs they are pulling back. No polling should be over-scrutinised. However, this is a volatile issue that all sides agree is crucial to election performance, and the female vote is being studied with intense interest by Tory strategists.

This week's budget will mainly be scrutinised through the familiar prisms of class and fairness – which income group bands benefit most, how much help is given on thresholds, what is said about the top rate of tax. But the gender effect may end up being more electorally significant. A largely male political and journalistic culture at Westminster sometimes ignores it – yet coalition policies are hardly gender-neutral.

This is because of where women work, the impact of public sector cuts on family services, and women's incomes. As a new Fawcett society report shows on Monday, it is women who are being worst hit. Very broadly, the public sector is more female than any other part of the economy – about 64% of public sector workers are women. The public sector pay freeze disproportionately affects women, mainly on lower incomes. Moves to break down national wage bargaining and pay people who live in less expensive areas lower salaries would disproportionately affect women. Public sector job cuts and hiring freezes disproportionately affect women. Female unemployment has been rising at twice the male rate, and is now at a 25-year high.

That, however, is just one side of it. The cuts affect women too, as users of services, including everything from Sure Start centres to the NHS, where they are in hospitals and clinics – often as parents and carers – more than men.

Add to that the effect of the sudden withdrawal of child benefit from the family incomes of those where one parent is earning £43,000 or more, the tax credit freeze, and the hugely negative impact of the NHS changes, and you see a cumulatively powerful message that cannot quite be dealt with by a Grazia spread.

Rather than brief the press that the health secretary, Andrew Lansley, is likely to be reshuffled soon (how does that help, once the bill is passed?), the government should bow to the last-ditch efforts being made in the Lords on Monday to amend or drop the NHS bill. The independent crossbench peer David Owen is trying to achieve this again on the arcane-sounding but important issue of the secrecy of the transitional risks register – basically, how much we are allowed to find out about the likely damage caused by these measures.

Meanwhile Labour peers, led by Baroness Thornton, will take the unusual step of trying to decline the third reading of the bill – to stop it reaching the statute book – on the grounds that it is unsupported by any democratic mandate, and widely opposed both by public and professional bodies. However, if these amendments fail, the coalition will continue to do itself damage with women voters over the NHS.

So what about the budget? Headline changes to the 50p rate or anything about tycoons and mansions are largely irrelevant to the argument. Of course, to the extent that income tax thresholds are raised more quickly, poorer working families will be helped – but that hardly adds up to a proper response.

In the past, notably during a visit to Sweden, Cameron has toyed with the idea of tax breaks to encourage working mothers to use domestic staff – half of the cost of cleaners, ironing services, gardeners, etc being made tax-deductible, perhaps. The idea is to free more well-qualified women to return to work. But I really don't think "tax breaks for your nanny", however popular in rural Oxfordshire, will quite cut the mustard as a big idea in lower- and middle-income Britain.

Something to help hard-pressed women who juggle caring, housework and children is clearly needed. But new research from the IPPR thinktank on gender roles in the home suggests childcare is the key issue. The most equal households are the ones without children at home, with a third of childless women under 40 sharing the household tasks equally with their husbands. Overall, however, 77% of married women do more housework than their husbands. The IPPR's Nick Pearce puts it simply: "Universal childcare, rather than tax relief for nannies or cleaners, is the best way forward for a family-friendly, more equal Britain."

Whatever you think of the Tories, politically stupid they ain't, and it would amaze me if a strategy for moving towards tax-deductible childcare was not part of the longer term Cameron-Osborne agenda. Even if it is too expensive just now, steps towards it (and a preparation for it), are entirely practical politics. It would not only provide some respite from the raft of women-unfriendly policies from the coalition so far but, by bringing more childcare workers on to the books, could actually help the formal economy.

Party politics aside, I very much hope they are thinking this way, because mothers and women carers can only benefit from a competition about this. Smart Tories know that to do well, they have at least to look family-friendly. Their big problem has been the effect of macroeconomic policy on families and women. Slow and tentative moves towards a society that is fairer towards women have been led and shaped by the state. The state brought in legislation, the state employs and serves. So it is hardly surprising that shrinking the state is likely to diminish equality too.

If you want to compensate in any way, you have to push far harder on new help for carers, and mothers who want to get back to work. Yes, women enjoy Grazia and gossip. But women are hard headed and count the money too. Particularly for poorer paid women, these have been a very hard couple of years, and the gender effect of the budget's measures on Wednesday deserves a lot more scrutiny than tycoon taxes.

Twitter: @jackieashley

• This article was amended on 19 March 2012. The original referred to David Owen as a Lib-Dem peer. This has been corrected.