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The reports were from private Snapchat videos that emerged on social media last week showing 22-year-old Wright attending a party at which many guests were allegedly inhaling from balloons.
The club announced on Friday that her registration at the club had been cancelled, but added that the player would have access to their support channels if needed.
The Charlton Athletic statement reads: "The club were made aware of the video and quickly investigated the incident."
"As a club we are disappointed with the behaviour which doesn't represent the standards the team upholds.
"The player is remorseful and has now left the club.
"The club would like to make clear that although the behaviour is not acceptable, the players welfare remains important to us and she will still have access to the club's support channels if needed."
Wright joined up with the South London club in January of this year after a spell with Millwall.
The 22-year-old was the subject of an investigation at her old club after she allegedly filmed a friend putting the paws of a dog onto the steering wheel of a moving car.
Wright was condemned by road safety charity Brake, who told The Sun that her alleged recent actions were completely unacceptable.
They said: "Driving and alcohol simply do not mix,
"When you're behind the wheel you're in charge of a potentially lethal weapon which requires full concentration and zero impairment to do so safely.
"We urge all drivers to never drink any amount of alcohol and drive - and certainly never do both at the same time."
Sound advice, and a cause we can all get behind.
Charlton Women currently sit in 11th place in the Women's Championship and will return to action this weekend against Lewes, having lost their previous game against Sheffield United 1-0.
Durham currently sit atop the Women's Championship remaining unbeaten after eight games of the season so far.
The 2019/20 season was abandoned after the coronavirus pandemic rendered the fixtures impossible to fulfil.
Aston Villa, who were six points clear at the top of the table and unbeaten in the fledgling season, were named champions when the league was ended prematurely on 25 May.
Charlton avoided relegation last season after results were named null and void.
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