The Manchester United striker will miss Tuesday's crucial clash against Ukraine after picking up a yellow card to go with his two goals at Wembley
The Manchester United forward scored twice at Wembley on Friday night but received his second yellow card of qualifying in the first half for kicking the ball away, leaving the Three Lions desperately short of strikers, with Wayne Rooney injured and Daniel Sturridge struggling with a thigh problem.
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"The win was overshadowed, especially coming at a time when we're short of front players anyway," Hodgson told the press. "It doesn't get much more unfortunate than that but we're getting used to dealing with it.
"We were shaken, if not stirred, by Welbeck's booking. I don't think there are too many people in the game who would say that's a yellow card - it was a 40-yard sprint and a first-time shot just after the whistle.
"He's got a yellow card, he's out of the game and we'll live with it. Did I think it's a correct decision? No I didn't."
Sturridge will join up with the rest of the England squad on Saturday night but Hodgson fears that the Liverpool forward has only a slim chance of being fit to face Ukraine.
"There's no reason why we should give up hope," he added. "It's not impossible but it's definitely no more than 50-50 - and I fear it's worse than that."