David Cameron has vowed to press on with controversial health reforms, insisting he will continue to champion radical and sometimes unpopular policies after the departure of his right-hand man, Steve Hilton.

However, in a sign of irritation with Andrew Lansley's failure to make the case for change in the NHS, the prime minister pointedly neglected to mention the health secretary by name in an address to party supporters.

Cameron told the Tory spring conference in London that his party would not be able to deliver a better and fairer country by ducking tough choices – such as those on welfare, the future of the NHS reform and the deficit.

"We won't get there by playing safe," he said. "To be strong and fair we've got to be tough and bold."

On the NHS, the prime minister said he wanted Conservatives to feel "proud" of the "tough decisions" to put GPs in charge of commissioning services and open the health service up to more private sector involvement.

"I know it has been a difficult journey," he said. "We're not reforming the NHS because it's easy. Let alone popular. We're doing it because it is right and necessary. In fact, it is more than that. It is unavoidable and it is urgent." He said he was prepared to "take a hit" in terms of popularity for the sake of health reforms that were right.

But while Cameron went out of his way to single out the education secretary, Michael Gove, for praise for his drive to improve education, he declined to name-check Lansley in a clear sign the health secretary is losing favour. It is now widely expected that Lansley will be removed from his post in a reshuffle after the summer holidays. Amid the furore over the reforms, No 10 has decided to take closer control over health policy, including the future of social care for the elderly, which will be the subject of a much delayed white paper in the spring or early summer.

Cameron was speaking the day after his friend and closest aide, Steve Hilton, announced he was quitting Downing Street to take a year's unpaid leave, joining his family in California. Hilton's move has caused unease in Tory circles, with many MPs voicing private concerns that Cameron has lost his most radical voice inside No 10, and will be more prone to compromise with the Liberal Democrats.

The prime minister said it was right to go ahead with plans for a high-speed rail link between London and the north that will carve up key Tory heartlands – a comment that prompted dissent from the audience.

On Monday, Labour will put pressure on Cameron and the chancellor, George Osborne, to reconsider reforms to tax credits and child benefits ahead of the budget on 21 March.

Labour says the government should cancel changes coming into force in April that will see up to 200,000 working parents lose almost £4,000 a year in working tax credits. Labour is also calling for the government to review proposed changes to child benefit which would mean that a family with two earners each bringing in £42,000, a total of £84,000, would keep all their child benefit, while a single-earner family on £43,000 would lose it.

Ed Balls, the shadow chancellor, will say during an opposition day debate that dumping the reforms could be funded by closing a stamp duty loophole on properties over £1m which is costing the Treasury hundreds of millions of pounds.