Since taking over as chairman of the Football Association in July, Dyke has made improving the playing resources available to the national team his priority, setting up a commission to look into the future of English football.
In a speech in September he raised concerns over the ease with which non-European players have been given work permits in the past.
Stoke were confident United States striker Juan Agudelo would be granted a permit when manager Hughes and chief executive Tony Scholes travelled to the hearing on Wednesday, but the 20-year-old's case was rejected with no right to appeal.
Agudelo had signed a pre-contract agreement and had been due to join the Potters in January when his deal with Major League Soccer side New League Revolution ran out.
"We are really disappointed with the decision," said Hughes, who also expressed his frustration that the reasons for the rejection were not given.
"If I'm honest, I think there's lesser players been allowed in at previous work permit applications than Juan Agudelo. I just think maybe the goalposts have been changed somewhat.
"Obviously there was a statement by Mr Greg Dyke in terms of what he felt was lacking in the game and maybe the committee has taken that on board. Unfortunately we were the first work permit hearing after those statements.
"You look at the criteria and he hadn't played as many games as they need to but other factors have to be taken into consideration - obviously his age and his injuries, which stopped him being selected - so there were reasons why he hadn't played the full complement of international matches.
"I felt that should have been taken into consideration and his ability as a player and the impact that he would have had for us should have been factored in, but it didn't seem to register.
"He's a very focused and determined young man, he wanted to progress his career, and I felt and he felt this was the perfect club for him. No doubt somebody in Europe will pick him up and he'll be a sensation somewhere else unfortunately."
The decision forces a rethink by Hughes on his plans for the January transfer window, but a striker remains the priority.
He said: "We're still in the market for other players, the focus hasn't all been on one player. But January isn't easy to navigate, so we'll just have to hope we get a little bit of luck.
"We've been looking for another striker to complement the ones we have since the summer, that's not a secret, so we continue to look in those areas."
Stoke face a resurgent Sunderland on Saturday looking for their first victory in eight Premier League matches.
Hughes has been encouraged by performances, with their most recent outing a 3-3 draw at Swansea in which they had led 2-0 before salvaging a late point.
The Potters sit above the relegation zone only on goal difference, and Hughes said: "We've done okay.
"We've picked up decent draws away at Swansea and at home to Southampton, who are flying, so we may look back at those results at the end of the season and think that they were decent results. At the moment we're probably thinking they were missed opportunities.
"It's important from our point of view that we start converting our performances - we need to start putting wins on the board. Draws keep you where you are in the Premier League whereas wins move you up the table, and that's what we want to do."
Sunderland's upturn in form has come since Gus Poyet replaced Paolo Di Canio in the hotseat, with wins over Newcastle and Manchester City lifting them off the bottom of the table.
Hughes was a team-mate of Poyet at Chelsea, and he said: "Gus is a great guy and a very infectious personality, so it's no surprise that he's gone in there and there's been a reaction.
"Gus wants his teams to get the ball down and play. They've had two really good home performances, maybe away from home they haven't been quite as accomplished.
"But it's up to us to make sure we disappoint them at the weekend because we're a good side and we've got the capability of beating Sunderland."