The Wales international stated to head coach Kenny Jackett 24 hours before the 1-0 defeat at Priestfield that he did not want to feature, coming after first-choice stopper Carl Ikeme was ruled out for up to six weeks with a broken hand.
But Hennessey, heavily linked with a January move to Premier League Crystal Palace, has since admitted to Jackett that his behaviour was incorrect and has taken full responsibility.
"Wayne has apologised this morning (Monday) and feels he was in the wrong and made the wrong decision," Jackett told the club's official website.
"With hindsight he wishes he hadn't said what he said and I am sure he will speak for himself but to me he has apologised and said it was wrong.
"In terms of the overall picture we haven't heard from any clubs or had any bids in yet. We don't know whether there is anything imminent but we are not close to anything as we haven't had a bid.
"In terms of Wayne he will carry on as a Wolves player until something changes - we haven't had any calls as yet and he will carry on as per usual.
"There will be some internal discipline and once that has been dealt with we will move on and put it down to an error of judgement.
"He is available for selection and will have no problems going forward as a Wolves player."
Meanwhile, Palace manager Tony Pulis is remaining tight-lipped over reports of a January move for Hennessey.
Pulis admits he is keen to bolster his ranks this month as the Eagles aim to preserve their Premier League status, although he is not the only boss to be linked with a potential swoop for the Wanderers 'keeper.
The 26-year-old is fit again after spending three months on loan at Yeovil having missed all of last season following two cruciate ligament injuries.
Asked about Hennessey following Palace's FA Cup win at West Brom on Saturday, Pulis said: "There are lots of reports saying he is going to West Ham and going here and going there.
"I think we will be linked with almost everybody over the next couple of weeks...especially on Sky Sports News."