One family has described the scorched grass as disrespectful.
It said weedkiller was only used where necessary.
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The issue was raised this week by one family said they saw no evidence of weedkiller being used at another church graveyard in town or at the war memorial gardens.
The issue was first noticed during lockdown.
A relative, who didn’t want to be named, said: “I noticed the grass had sprayed with some very strong weedkiller
“Six inches of grass around the grave stones had been completely burnt.“I was angry, but because it was lockdown I appreciated that things were different.
“Now it has happened again, and it seems to only be at Hayfield.
“Apart from destroying the environment, it is disrespectful to to do this in cemetery.”
The woman said she had raised the issue with women in the cemetery and also written to bereavement services, sending them photos of the state of the grass.
She also visited Abbotshall Church’s cemetery and said it didn’t seem to have the same issue with weedkiller on grass verges.
“We need a bit of a shift in mindset,” she said. “I’lm not looking for the grass to be perfect. Generally the cemetery looks okay, but they coud strim round the edges rather than use weedkiller.
“In parks the grass is longer to encourage wildlife - if they can do it there, why spray cemeteries with weedkiller?
“Instead of perfect green grass, we can accept something which might be a bit longer and good for the environment.”
Thomas Graham, lead officer, said the department had apologised to the family for any distress caused.
He added: "Our teams are all trained in grounds maintenance. We use a variety of methods to keep our cemeteries tidy including grass cutting, strimming and weedkiller where it's necessary.
"We work very hard to keep our cemeteries looking their best. We understand how important it is for visitors to have a peaceful and well maintained space."