"Just going in a different direction," Woodson said. "Nothing against Kidd. He's still gonna play. ... Just changing it up a little bit."
Kidd has struggled mightily with his shot, making just seven of his last 49 attempts from beyond the arc. Prior to Friday's game, Kidd had shot just 19.6 percent from the floor in his past 10 games.
It is unclear how long Woodson intends to keep Kidd on the bench. He said after Friday's game that he'd evaluate the situation on a game-to-game basis.
Before tipoff of Friday's game, Woodson said he'd inserted White into the starting five because of his defense.
"Just from a defensive standpoint, we're going against (Washington's Martell) Webster," Woodson said. "It's a better matchup for Webster."
Webster had 19 points in Washington's 106-96 win over the Knicks on Feb. 8. He had just nine in 37 minutes on Friday.
Kidd has struggled at times on the defensive end against bigger, quicker shooting guards.
Woodson said he'd like to see Kidd handle the ball more, one reason why he'd sent him to the bench.
"It's what he's done all his career," Woodson said, though Kidd has thrived as an off-guard more recently.
Kidd will be the primary ball handler in the Knicks' second unit for the time being. That job formerly belonged to Pablo Prigioni.
"He's a point guard, and I've got to put him in a position to be more successful," Woodson said of Kidd, who is under contract through 2014-15. "I don't think I've done a good job in that area. ... I want him more with the ball in his hands to run this team."
It worked on Friday as Kidd had four points, five assists and three rebounds in 28 minutes. He helped facilitate for the second unit and helped the Knicks close the Wizards out in the fourth quarter.
After the game, Kidd said he has no problem with his new role.
"Whatever it takes," the 39-year-old said. "Coach and I talked about it and I was in agreement with whatever he needs me to do."
Woodson said he intends to use lineups in which Kidd and starting point guard Raymond Felton play alongside one another. He said he would find minutes for Prigioni and would also try to play Kidd alongside Prigioni.
"They're going to share the task of running the team, which is OK, I think," Woodson said. "I'm going to try to fit all of them in. It's a lot of players."
Kidd, 39, began the season shooting well, particularly from beyond the arc. In the Knicks' first 20 games, he was hitting 50 percent of his 3-point attempts. But Kidd began to tail off after Felton went down with a fractured right pinkie.
Kidd logged 30-plus minutes in eight of the 12 games Felton missed in December and January.
Once Felton returned, Kidd's production decreased.
He'd been giving the Knicks' minimal offense in February, failing to score in four of his last seven games. Kidd totaled three points or fewer in eight of 11 games in February.
He was unavailable for comment prior to the game.
In limited minutes, White has shown that he can defend on the perimeter.
Aside from participating in the dunk contest during All-Star Weekend, though, White hasn't been very active lately. He's played 10 minutes or fewer in 16 of the Knicks' last 18 games. He did not appear at all in seven of those 18 games.
Ian Begley is a regular contributor to ESPNNewYork.com.