"To deny the request" means to refuse or reject someone's appeal or plea for something. It means saying "no" or declining their request. Full definition
However, the judge denied my request for jury, and even went so far as to say that no such right to trial by jury existed for civil cases. (law.stackexchange.com)
She also plans to deny requests from other county lawmakers to put their names on signs at taxpayer - funded projects. (newsday.com)
The court denied his request to dismiss the case and ordered the seizure of the undisclosed assets for liquidation and distribution to his creditors. (downriverbankruptcy.com)