When deciding to approve or deny a time off request you should look at your policy, past precedent, and the circumstances of the particular situation. If the request for extended time off is actually for a health-related reason which may be protected under laws such as ADA or state disability, you may need to approve the time off.
However, if the employee does not provide any protected reason, you do not have to offer more than your policy or what you have done in the past in a similar situation. If your policy is to approve two weeks, you can explain that you are not able to guarantee they will have a job to return to if they take more time.
You also should consider the impacts if you let the employee effectively resign. Will you be able to fill that position and get a new employee up to speed in that period of time? Or would it be easier to just manage for a couple of extra weeks?
Of course, whatever you do, you will be setting a precedent for future employees who may want to stretch their time off for personal reasons.
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