Mutch joined Crystal Palace on Thursday for a fee believed to be around £4.75million, having made only 11 appearances since moving to Loftus Road last summer.
Rangers had been looking primarily at loan deals but the Mutch transfer means Redknapp may be able to spend before the January transfer window shuts on Monday night.
"We had an offer for him and it was a money back type of deal," Redknapp said.
"He's a good lad and I liked him a lot but he hadn't really got up and running here.
"There's plenty of potential in him but we're looking to get one or two in so we had to try to raise some money if we could.
"We're just struggling to bring anyone in, that's the problem."
Redknapp ruled out signing a number of players reportedly linked with the club, including Yann M'Vila, Rudy Gestede, Aaron Lennon, Dani Osvaldo, Bacary Sagna and Emmanuel Adebayor.
Rangers had made an enquiry for Sagna but made no progress with the French defender while Redknapp said any deal for Adebayor would be too expensive.
"He's at Tottenham and I don't see it happening," Redknapp said.
"Adebayor is an excellent player
You don't play at Arsenal, Real Madrid, Manchester City and Tottenham if you're not a good player.
"But I don't see him leaving Tottenham, it's an expensive deal and we're looking at a couple of loans.
"If we can find them we'll do them but we need to get them over the line and at the moment we're not close to anybody.
"The chairman is trying for his life and Les Ferdinand is working on that side of it as well, but it's not easy."
Rangers will train at England's Burton training base at St George's Park on Friday as they look to end a miserable run of results away from home against Stoke.
Reknapp's side have lost all 10 of their league games on road this season.
"It shouldn't be a psychological factor," Redknapp said.
"They're passionate fans at Stoke and it's a difficult place to go but we have to go away and pick up some points
It needs to start sooner than later.
"We've done little bits in training but it's about players
You can change what you eat for dinner, whether you go on the coach or the train, but it's about players and how they perform."
Source : PA
Source: PA