Could a cash strapped government in a depression/repression simply print money to fund a large project (eg Manhattan Project)?

The philosophy of John Maynard Keyenes, which we call Keynesianism, was that government spending could always pull us out of a recession, in fact it's the only way. Republicans pretend not to be Keynesians but they also use govt. spending to stimulate the economy, just they spend the money on different things.

Actually what Keynes recommended was that govt. put money aside during good years, that it deliberately run a budget surplus during good times so it would have extra money to spend during bad times. And we sure haven't done a good on that half of the plan.

Yes Obama did that to pump trillions into Fannie and Freddy

It's not as simple as that.

What you are suggesting causes recession and depressions, the destruction of industries, and unemployment, poverty and homelessness, which can't necessarily be fixed later on by destroying money that was printed.

“What Has Government Done to Our Money?” by Murray Rothbard
http://mises.org/books/whathasgovernmentdone.pdf

“The Mystery of Banking” by Murray Rothbard
http://mises.org/Books/mysteryofbanking.pdf