The fund was established to support private sector jobs and growth in places that rely on the public sector.
But a National Audit Office report says there was scope to have generated more than the 41,000 jobs expected relative to the amount of grant awarded.
The fund has not optimised value for money because a significant proportion of the £1.4bn was allocated to projects that offer relatively few jobs for the money invested, the report says.
The expected cost per job varies considerably between projects, it says, from under £4,000 per job to over £200,000 per job.
Applying tighter controls over the value for money offered by individual bids and then allocating funding across more bidding rounds could have created thousands more jobs from the same resources.
The full report can be downloaded by clicking here.
"The Regional Growth Fund, which was set up to support growth and jobs in areas that rely on the public sector, could result in 41,000 additional jobs," said Amyas Morse, head of the National Audit Office.
"However, some of the funding was allocated to projects that offered relatively few jobs for the money invested.
"To achieve better value for money from the further £1 billion now available, the government should develop more challenging targets for the number of jobs projects should generate relative to their cost."
Rigorous evaluation will be required to quantify precisely the fund's overall employment impact, says the report.
More than two thirds (28,000) of the 41,000 additional jobs are expected to be delivered indirectly, for example through knock-on effects in companies' supply chains or the wider economy.
The average project will last at least seven years. But it is unclear how much of the fund's boost to the private sector will be sustained in the longer term.
It has also taken longer than expected to turn conditional offers of grants for projects into final offers.
Therefore, despite the government's intention to get projects up and running quickly, the report says only around a third have so far received final offers of funding.