Manager Jo Rowe said the centre was affected more than others as it provided after-hours care, which was the most convenient time for many parents to take children to the doctor.
Since July this year, doctors' visits for under-13s have been funded by the Government, but it is capped at the equivalent of two visits per child per year.
''Utilisation rates for under-13s have grown exponentially - certainly greater than what we expected them to,'' Mrs Rowe said.
Attendance for the age group had increased by at least 20%, but practice data would need to be analysed to get a precise figure.
Practices would have no option but to reinstate co-payments if funding ran out.
''It would be really sad if it went.
''How do you stop people coming in for a third visit and using that funding?''If they're going to cap the amount that's in that fund, practices simply are not going to be in a position to continue to offer that service [without co-payment],'' Mrs Rowe said.
Last month, the Otago Daily Times reported the South's primary health organisation, WellSouth Primary Health Network, had had early reports of much-higher-than-expected uptake.
Yesterday, WellSouth chief executive Ian Macara said some practices might withdraw from the scheme.
If necessary, WellSouth would advocate for more funding, he said.
''In other words: `Give us more money because the presentations are way higher than what was being modelled nationally.
''I know this Government is committed to providing better access for that age group, so that's a good sign.''
Feedback from around New Zealand suggested uptake was highest in cities.
In the South, it meant the problem might be confined to Dunedin and Invercargill: ''That's what we're hoping. We don't know until we see the data''.
Of 83 practices in the South, 81 were in the scheme.
A Ministry of Health spokeswoman said higher uptake was to be expected at first.
When fees were removed from the under-6 age group a few years ago, uptake increased before levelling off, the spokeswoman said.
The flu season might also be playing a part, she said.
''Funding is based on two visits per year per child aged 6 to 12 years.
''The national average utilisation rate for the 6- to 12-year-old age group is 1.75 daytime visits per year.''
The scheme would be reviewed if the numbers continued to be higher than expected, the spokeswoman said.
