The government has come under immediate pressure from its own MPs over a £1.6bn funding boost for deprived English towns, as it emerged the guaranteed spending in some regions would amount to as little as 58p per person per year.

The so-called Stronger Towns Fund, announced overnight, is billed as a chance to assist left-behind communities, but has been disparaged as a crude attempt to bribe Labour MPs representing these areas to potentially back Theresa May’s Brexit deal.

The money, to be spent over seven years, includes £1bn pre-allocated to various English regions, ranging from £281m for the north-west to £33m for the south-west. The other £600m will be available to any region via a bidding process.

The communities secretary, James Brokenshire, stressed that the money was not contingent on the deal passing. “This funding is there regardless of the outcome, but obviously we want to see a deal happening. We believe that is what is in the best interests of our country,” he told BBC Radio 4.

But answering ministerial questions later in the Commons, Brokenshire came under pressure from the Conservative MP Henry Smith, who said his Crawley constituency had some of the most deprived wards in the south-east of England.

“What I don’t want to hear from the frontbench is that we’re somehow in the B list, where we can bid for that funding. This funding is needed now,” Smith said.

Brokenshire said he would respond more fully in a Commons statement on the fund later on Monday, adding: “But it is a £1.6bn fund with an element that is competitive, and I would certainly encourage people to bid into that.”

Both the perceived bribe and the scale of the money made available have seemingly not impressed the Labour MPs at whom it was aimed, with Gareth Snell saying the entire allocation was less than the total cuts faced by his local council, Stoke-on-Trent, over the past four years.

There is also disquiet among some Conservatives at the agreed allocation of the £1bn element, and the big regional variations this includes. The north-east will receive £105m, or nearly £40 per head of population, which amounts to £5.70 a head per year over the seven years of the fund.

The per-person-per-year sum is also above £5 for the north-west, Yorkshire and Humber, and East Midlands. However, it falls to £3.30 for the east Midlands, and 85p per head of population a year in the south-west, which has a number of deprived areas.

The lowest spend is in south-east England and the east of England, which will have 58p allocated per person per year.

Speaking at departmental questions, the shadow communities secretary, Andrew Gwynne, said that even the £281m allocated for the north-west was notably less than the £1.5bn reduction to core council funding under austerity.

Gywnne told MPs: “So after nine years of this government’s slash-and-burn approach to deprived areas, the secretary of state has announced a new fund for our left-behind towns.”

He told Brokenshire: “Does he understand why members across the house feel disappointed and patronised by his announcement today?”

Earlier, May’s spokesman insisted the money would be new spending, and not taken from the similar shared prosperity fund, a previously announced scheme to reduce regional inequalities and replace EU funds playing a similar role.

Details of the shared prosperity fund, which had been due earlier this year, would come shortly, the spokesman said.

He added: “The prime minister has said, right back to the comment she made on the steps of Downing Street, that for too long in our community prosperity has been unfairly spread. Our economy has worked well for some places, but we want to ensure it works for all communities.”

Earlier, the Joseph Rowntree Foundation poverty thinktank said the new town fund “must not mean dipping into the shared prosperity fund, a manifesto commitment made to towns and cities using money repatriated from the EU”.

The consultation on this fund was “long overdue and has left towns and cities facing uncertainty as the Brexit withdrawal process saps Whitehall attention”, it added.