Red Sox starting pitcher Clay Buchholz tosses a pitch during the first inning of last night’s game against Kansas City.

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AP / Ed Zurga

KANSAS CITY, Mo. — Clay Buchholz couldn’t figure out how to get a big out in the third inning. For most of the night, his teammates couldn’t deliver a big hit. Taken together, those two failings provided a rather predictable result.

Stranding an even dozen — including three in the ninth inning — the Red Sox saw two modest three-game winning streaks come to end: one overall and another on the road as they dropped a frustrating 4-3 decision to the Kansas City Royals.

The Sox had their chances against closer Joakim Soria. A bizarre collision between Kansas City third baseman Alex Gordon and shortstop Tony Pena Jr. enabled Jason Bay to reach on a run-scoring infield hit and bring the Sox to within a run.

Sean Casey then lashed a liner that looked for a minute to have a chance to fall in. But right fielder Mark Teahen gloved it, and the Sox and extended their futility with runners on base (3-for-16 after the first inning) for the night.

“I really thought like I hit into the corner,” said Casey, “but [Teahen] was right where I hit it.”

“We gave ourselves a chance,” lamented manager Terry Francona. “Even 0-and-2, Casey took a great swing. But we had dug ourselves too big a hole.”

It was that kind of night for the Sox, who have yet to win more than three in a row away from Fenway all season.

The Sox’ offense was silent between the first and last innings. A walk to J.D. Drew, a double by Dustin Pedroia, an infield out by David Ortiz and a run-scoring double by Kevin Youkilis jump-started the Sox in the first to a 2-0 edge over Kansas City starter Gil Meche.

But though Meche labored, needing 109 pitches to get through six, they couldn’t use his five walks to their advantage.

In the eighth, after singles from Casey and Jed Lowrie gave the Sox runners at the corners, the Royals seemed to play into the Sox hands by bringing in lefty Ron Mahay to turn around Jason Varitek to the right side, where he’s been far more effective. But Varitek, in a nine-pitch battle, swung at a pitch out of the zone for strike three.

In the meantime, Buchholz extended his winless streak to seven games. He is 0-4 with an ERA of 6.59 in five starts since his stint on the DL and rehab assignment to Pawtucket.

He hasn’t won since May 2 and his spot in the rotation could be in jeopardy when Bartolo Colon, who begins a rehab assignment tonight for Pawtucket, builds up sufficient arm strength to return.

As has been the case often with Buchholz, one bad inning did him in. He allowed a solo homer to Gordon in the second to cut the 2-0 lead in half, but the real problem came in the third, when the Royals sent nine men to the plate and scored three times.

Mike Aviles reached on an infield single that Pedroia couldn’t field cleanly, and it went downhill from there. Esteban German hit a rocket at Lowrie, which exploded right in front of the rookie, putting two on.

“That was a tough hop,” said a sympathetic Buchholz. “It hit about three feet in front of Jed and took off. That’s 50-50 for anyone to handle. I can’t fault him for not making that play.”

Buchholz didn’t help matters with a wild pitch to send both runners into scoring position, and Teahen’s single to left scored Aviles. A walk to Jose Guillen loaded the bases, and Billy Butler then produced a sacrifice to right, plating German.

A walk [to Gordon] and a hit batsman [John Buck] produced another run.

“I hit Buck with a two-seamer that was a purpose pitch,” Buchholz explained. “I wasn’t even trying to throw a strike; it just happened to hit him.”

Buchholz got out of the third, then allowed just two hits in his final three innings, but the damage had been done. In every one of his five starts since being recalled, Buchholz has had an inning in which he’s allowed multiple runs.

“I don’t know why it’s been like that all year,” he said. “It seems like I need to make that one pitch and I don’t; it feels like it’s always an inch away. I don’t have a very good explanation for it. I feel like sometimes I’m trying to make pitches better than I need to.

“It’s been a challenge for me.”

“I think we saw after that (inning), he settled down and pitched with confidence,” said Francona of his young starter. “He needs to limit the damage. He just didn’t make the pitches. You can almost see him trying to will it in there.

“Like we said, he’s still a work in progress. But I do like what we saw with those three good innings (after the third).”

smcadam@projo.com