We should add API which will return RDF (probably TTL) data in the following format:
wds:Q43$66E13CA3-37CB-4B14-9F53-311D9C91D822 wikibase:hasViolationForConstraint wds:P17$a02094d5-46fc-bb86-81d1-21cfea688dfe . wds:q43$43124bad-4210-425f-5a0f-56e86c922055 wikibase:hasViolationForConstraint wds:P17$f2713cbe-499d-36e8-0a3a-c275a356b659 .
etc. etc. for each constraint violation. The left side is the statement violating the constraint, the right side is the statement on property that defines the constraint.
So far the easiest way seems to be to add action API, e.g.: https://wikidata-constraints.wmflabs.org/index.php/Item:Q5?action=constraintsrdf