I'm looking for a phrase that is like 'special dispensation', but I know that's not the phrase I'm looking for. I'm trying to say that work had to be done on object X that was different, and unique, for objects Y, Z and A.
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Depending on the context, there might be a term that's more common in one line of work or another, but a core word to this is "case," as in special case, unique case, or edge case. Each one carries a sense of variation from some common or expected procedure.
Speaking of which, terms like "variation" or "exception" might also work. Variation often applies in policy-oriented contexts. For example, a fire safety code may require all hallways narrower than a certain width to be clear from wall to wall. Some lab-oriented building that needs to place equipment in the hall has to apply for a variance.
For a variance that exists due to age, 'grandfathered' may be what you are looking for.