Can I use a minor variance application to legalize a structure that was built without proper approval?

Yes, this is a common use of the minor variance process, sometimes referred to informally as a retroactive or after-the-fact application, where an owner asks the committee of adjustment to approve the specific zoning non-compliance an existing, already-built structure represents, rather than seeking approval before construction the way the process is normally used. The committee applies the same established four-part test to this kind of application as it would to any other minor variance request.

The fact that the structure already exists doesn't change the legal test being applied, and it doesn't guarantee approval; the committee still has to be satisfied the variance is minor, desirable, and consistent with the general intent of the zoning bylaw and official plan, based on the situation as it actually is. In practice, an existing structure that otherwise fits comfortably with the neighbourhood may sometimes be viewed favourably, but a poorly sited or genuinely problematic structure faces the same risk of refusal as it would if the application had been made before construction, potentially leading to an order requiring the non-compliant portion to be altered or removed. Getting a realistic assessment before applying, rather than after enforcement begins, is the better approach.