Kei Kamara of Sporting Kansas City kicks the ball for a first-half goal against D.C. United. (G. Newman Lowrance/GETTY IMAGES)

D.C. United embarked on a two-match swing through the Midwest four days ago beaming with confidence and cohesion, but after a 1-0 loss to Sporting Kansas City on Sunday night, the club turned back home with just one point and a greater sense of urgency heading into the final 10 games.

United (7-7-10, 31 points) lost on the road for the first time in almost four months and missed another opportunity to crack the top three in MLS’s convoluted Eastern Conference.

Playing its third match in nine days, United showed signs of fatigue and struggled to keep pace with Kansas City’s swift and diverse attack. Kei Kamara scored from 18 yards in the 19th minute, but the margin seemed wider. Other than Blake Brettschneider’s bid late in the first half, United was harmless.

“When physically a team is shot, sometimes the mind goes a little bit,” United Coach Ben Olsen said. “We knew what to expect from them and played into their hands. They come at you, they punch you right in the face at the beginning, and if you’re not prepared for it, they usually get a goal.

A poor playing surface, ravaged by the Farm Aid concert last weekend, didn’t help matters, but Kansas City’s three-forward formation navigated with the field with superior prowess. United was a step behind throughout the evening, lacked energy and direction, and was fortunate to trail by just one goal at halftime.

With 15,352 watching, Kansas City (9-7-9, 36 points) improved to 6-1-4 at freshly minted Livestrong Sporting Park and leapt two clubs to claim second place behind the Columbus Crew (11-7-7, 40).

“I don’t know if it’s mental fatigue or physical fatigue, but unfortunately we weren’t on as a team,” Brettschneider said.

The biggest pregame intrigue surrounded United’s goalkeeping situation. With Bill Hamid sidelined with a strained hamstring, Olsen chose rookie Joe Willis over veteran Steve Cronin for the second time in three matches.

There were other notable changes. Leading scorer Charlie Davies, ineffective in recent weeks while recovering from a sore knee, was not in uniform. Dwayne De Rosario, the most influential figure in the attack, needed a break and began the evening on the bench.

“Dwayne is not 22 anymore,” Olsen said of De Rosario, 33, who had played all but six minutes in eight appearances since being acquired from New York. “He’s been putting us on his back for a couple of weeks and I wanted to be smart with him.”

Kansas City dictated terms against a United side that was ragged and out of rhythm — a repeat of the creaky start during a 1-1 draw at lowly Chicago on Thursday.

Two minutes after Omar Bravo’s 10-yard header skipped off the crossbar, Kansas City went ahead. United’s Daniel Woolard headed away Bravo’s cross, but Kamara settled the ball at the top of the box, side-stepped Dejan Jakovic and cracked a pulsating right-footed drive into the lower left corner for his seventh goal of the season. Willis never had a chance.

United stirred to life late in the half, but goalkeeper Jimmy Nielsen’s sensational reaction save thwarted Brettschneider’s angled effort from 12 yards.

United was in desperate need of change, so after the break, Olsen inserted De Rosario, who brought composure on the ball and a menacing quality. In the 52nd minute, he curled a 22-yarder over the crossbar.

Jakovic cleaned up Woolard’s wayward touch before Teal Bunbury could pounce and cut off Kamara’s dangerous cross. Willis pushed Graham Zusi’s 25-yard rocket over the bar. United was living dangerously, but with De Rosario involved, the outcome remained in the balance.

In the 78th minute, De Rosario was a step from connecting with Andy Najar’s deflected cross — the last authentic glimmer of hope.

“The more the game went on, they kept pressing and we got more tired,” Willis said. “It just wasn’t a good situation.”

United notes: The club’s next five matches are against Western Conference teams, starting Saturday with the Portland Timbers at RFK Stadium. United is 4-2-6 against Western foes. . . . United’s away unbeaten streak ended at seven (3-0-4).