The 46-year-old's application, listed to be heard at the Old Bailey court in London, was abandoned after News UK - the rebranded Rupert Murdoch firm - said it would not be applying for any costs in relation to the trial.
The publisher's decision also means other defendants in the phone hack trial who were acquitted have also dropped their cost claims, saving the British taxpayer millions.
"It is for that reason that News UK have indicated it did not feel willing to engage in an exercise addressing these issues," Smith said.
According to the 'Guardian', News UK's decision not to reclaim costs, although expensive, means it avoids a potentially damaging and protracted scrutiny of its stance during and before the trial.
Charlie Brooks, husband of the former chief executive of Rebekah Brooks, is seeking ballpark costs of 600,000 pounds.
Stuart Kuttner, former managing editor of the now defunct 'News of the World', is seeking 135,000 pounds of costs incurred before News UK indemnified him in January 2013.
Under UK law, anyone found not guilty at the end of a trial is entitled to make an application to be reimbursed for the court costs incurred.
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