MPs will be able to claim extra expenses for accommodation, travel and staffing costs under new relaxed rules announced today which will cost the taxpayer up to £4 million extra a year.

The reforms come after MPs complained the new system was too  “bureaucratic” and “anti-family”.

From April, they will be able to increase their rental budget by £2,425 for every child they have — under 18 and in full-time education — to accommodate them at a second home. They will also be able to claim the cost of their children’s travel to and from their constituency.

Under the current system, MPs can only claim for the travel and accommodation costs of children under five.

The amount they can claim for staffing costs will go up from £110,000 to £115,000 a year.

And 31 more MPs will be able to claim for hotels or rent for a second home.

MPs will also be able to claim lump sum payments upfront and use taxpayer-funded credit cards with a £4,000 monthly limit for their expenses.

The changes have been made to calm the rage of scores of MPs from all three of the main political parties who have condemned the rules on expenses as too “complex”. The strict regime was imposed by Westminster’s new anti-sleaze watchdog after the 2009 expenses scandal.

Many MPs have complained they have been left seriously out of pocket by having to pay bills upfront from their personal accounts and claim the money back.

The new Independent Parliamentary Standards Authority had been warned by Prime Minister David Cameron to “change or be changed” by April 1.

Ipsa chairman Sir Ian Kennedy denied the body had been “bullied” into relaxing the rules.

He insisted there would be “no going back to the old ways” and said Ipsa had already delivered savings of £18m in its first 10 months.