Cook, whose propensity for verbal blunders has sometimes overshadowed the work he has done in driving a major transformation of the club during his three years in the job, said he was stepping down because of an error of judgment.
"Following the findings of the review, Chief Executive Officer Garry Cook has offered his resignation, which has been accepted with regret by the Board," the club said in a statement on its website.
Anthonia Onuoha, who acts as her son's agent, had emailed Cook saying that while she was "ravaged with cancer" it would not stop her negotiating on the player's behalf over a contract dispute he was having.
She received a reply, which seemed to be intended for the club's football administrator Brian Marwood, saying; "Brian... Ravaged with it!!...I don't know how you sleep at night. You used to be such a nice man when I worked with you at Nike. G"
"The club can confirm that there is foundation to Dr Onuoha's allegations and the chairman has written to apologise to Dr Onuoha for any distress caused," the statement said.
Cook has often found himself in the spotlight for ill-judged remarks but where many have been laughed off as gaffes, this latest episode has cost him his job with the world's richest club ahead of their Champions League debut.
In 2008, City's then owner Thaksin Shinawatra was charged with corruption but Cook defended the former Thai prime minister saying he was "a great guy to play golf with."
MAGNIFICENT ACHIEVEMENTS
He accused AC Milan of "bottling it" when City's attempts to lure Kaka to the club for 100 million pounds failed in 2009.
While inducting former City favourite Uwe Roesler into the club's hall of fame, Cook said he would like to welcome him to "the Manchester United" hall of fame.
Local media had reported this week that Cook had denied sending the email, saying that his account must have been hacked, but on Friday he spoke of making a mistake.
"I am privileged to have held my position at Manchester City Football Club and to have experienced the opportunities that it has presented," he said on the website.
"The privilege is in part offset however by the significant personal focus which has at times, detracted from the magnificent achievements of those working at the football club.
"It is that factor, together with my error of judgment in this matter that has prompted me to reach this decision, which I believe is in the best interests of the football club."
City, who ended a long trophy drought by winning the FA Cup last term, said director John MacBeath would fulfil the role temporarily before a permanent successor was appointed.
Chairman Khaldoon Al Mubarak thanked Cook for his contribution to the club, which has invested heavily in players and infrastructure since the takeover by Abu Dhabi billionaire Sheikh Mansour three years ago.
"His [Cook's] judgment in this matter should in no way lead
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