When the Otago Daily Times visited the cemetery on Thursday there were many large holes, some about 15cm in diameter.
CODC parks and recreation manager Mathew Begg confirmed it was the council that maintained the Clyde cemetery, which included mowing and spraying the plot surrounds, as well as undertaking interments and topping up graves as required for the 12 months after a burial, and again later if needed.
Begg said rabbit control was undertaken ''periodically'', but was ''proving to be a challenge to keep on top of'' at the Clyde cemetery.
A council contractor was going to inspect the cemetery last week, and ''liaise with council on a plan for ongoing control''.
When asked if the council was already working on a plan for rabbit control or if it had decided to do so after the woman's complaint about the Clyde cemetery, Begg said: ''We carry out work at cemeteries as required, with us already having done work at Cromwell and Ranfurly cemetery in previous years, and Clyde was on our list before we had talked to [the woman]''.
He said headstones and cemetery plots were the responsibility of the family, ''though it becomes an issue when the plots are historic with no descendants in the area to maintain the plots/headstone''.
The council's parks team leader was going to meet a contractor on site next week to look at the leaning headstone ''and how best to address this''.
All cemeteries maintained by the council were regularly inspected either by the council's contractor or council staff, Begg said.