A faction of disgruntled supporters believe Everton have stagnated under Kenwright due to a lack of investment and want him to sell his stake in the club.
Kenwright has long said he is willing to do that if he can find a suitable investor but there have been peaceful fan protests against his ownership this season.
Some fans also voiced their disquiet after last week's frustrating Barclays Premier League draw with Blackburn, but Moyes believes it is up to the team to put that right.
Moyes, whose side are 14th in the table, said: "The only negativity there is is because of how the team play. We have not played well enough.
"It is nothing to do with the board of directors or the chairman, it is to do with me and the team.
"We have got to do better on the pitch. If the team plays better it lifts the mood and people change.
"We need to perform better and win more often. If we do that it will change everything.
"Winning football games is always the answer so I will take responsibility for that."
Moyes has brought in Los Angeles Galaxy's Landon Donovan on loan and Darron Gibson from Manchester United this month but the club have still not paid a significant transfer fee for more than two years.
The other notable piece of business in the current transfer window so far is outgoing with Diniyar Bilyaletdinov poised to complete a move to Spartak Moscow.
A fee thought to be around £5million has been agreed for the 26-year-old Russian winger, who has failed to command a regular starting place since his £9million arrival from Lokomotiv Moscow in 2009.
Moyes said: "Billy has been here and maybe not quite reached what we wanted but I couldn't fault anything he did.
"I am disappointed because he was such a good professional and a great boy to work with.
"He was tremendously committed when he played for us and was really keen to be a success here.
"From that point of view I have no failings with him at all and we wish him well."
Moyes is now looking at whether he can still add further to his squad before the closure of the window next Tuesday.
He said: "We are asking around a little bit to see what we could maybe get but at this moment in time we have got nothing to report."
Moyes' immediate priority is Friday night's FA Cup fourth-round tie against Premier League rivals Fulham at Goodison Park.
The Scot said: "It is a tough game but it would have been tougher at Craven Cottage.
"They have brought some good players in but they already had some good players before that.
"It is a Premier League tie and that keeps the players' mentality levels up - the attitude level has to be right.
"They are up against players of very similar ability and they understand the games are tight.
"It is important to win. It is a last chance for us to win a cup competition this year and we want to get through to the next round."
Moyes must decide whether to continue with 20-year-old Shane Duffy in defence in the absence of injured Sylvain Distin and Phil Jagielka or reshuffle the back line by recalling fit-again Tony Hibbert.
Duffy, who had a loan spell at Scunthorpe earlier this season, impressed on his home league debut against Blackburn.
Moyes said: "Shane has done really well in the games and it is terrific for the boy.
"He will have a good career in the game and I hope it will be at Everton.
"If he plays as well as he has in the last two games we will all be more than happy."