The 63-year-old, who was in charge of the Hatters between October 2009 and March 2011, was appointed Pools chief on December 11.
He led the team in six games, with one win, before being moved into a 'senior role' last week.
However, Money has now decided to leave the club completely, as a statement on the club's official website, said: "Following discussions on Monday, Richard made it clear he didn’t want to burden the club with additional cost and offered to cancel his contract.
"It was therefore mutually agreed that Richard should be released from his contract.
"During his time here, Richard assisted in a number of player signings, which we believe will be of benefit to the club this season for which we are grateful."
Co-owner and chairman Raj Singh added: “Whilst we are disappointed, I would like to thank Richard for his honesty.
"I respect Richard’s integrity and the fact that he didn’t wish to further financially burden the club and wish him all the very best for the future.”