Q. When I see the phrase “contact at work, with discretion”, what does that mean exactly?
This is a phrase used when a person has a job but wants to be interviewed for a new job with a different company. It is saying “You can call me at work, but make sure my current employer doesn’t know you’re calling me about a possible new job so I don’t get fired.” If a potential new employer called you at your current job and told the person answering the phone, “I’m calling from Megatron Inc. about that job she applied for at our company,” that would be a complete lack of discretion.
As a job candidate, you might use the phrase “contact at work, with discretion” in your cover letter. You might also see it used by a potential employer on an application, which means the employer wants to know if they can call you at work but promises to not let on that they are calling about a possible new job.