The 66-year-old was attempting to urge his players to pass to the impressive Andros Townsend, although one player is alleged to have taken offence with supposed racist connotations.
Townsend himself insisted via Twitter that no offence was taken, with the Football Association announcing their backing over the fallout - and Hodgson says insists the rest of the England squad support him.
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"We have just had a successful period and, although I wouldn’t suggest we intend to rest on our laurels, I think we have earned the right to enjoy the fruits of our labours. Instead we get this."
Hodgson apologised for the joke following the media coverage, but maintained that no offence was taken by the Tottenham man.
"I would like to apologise if any offence has been caused by what I said at half-time," he stated on Thursday.
"There was absolutely no intention on my part to say anything inappropriate. I made this clear straight away to Andros in the dressing room.
"I also spoke to Andros again on Wednesday. He has assured me and the FA he did not take any offence and understood the point I was making in the manner I intended."