Chorus says work is underway to get its services back up and running.
In a statement, the telecommunications provider said issues on its network had been caused by power outages in the Auckland region caused by the storm late Tuesday night and early Wednesday morning.
The storm, the result of a Category 2 cyclone, pummelled Auckland with gusts of up to 140km/h.
Power outages since then meant some of its cabinets and exchanges had been running on battery back-up systems or generators and were now running flat.
"Due to the length of time power has been out in some areas, some of the back-up batteries have gone flat. Chorus is working closely with its service partners, as well as Vector and Counties Power to prioritise restoration," it said.
"Fuel runs are planned to keep generators working, and additional generators and spare batteries are also being sourced."
About 100 of its power cabinets were without service in Auckland, impacting 11,000 customers.
"The cabinets are in areas where Vector and Counties Power are still working to reconnect power."
Chorus said there had also been an increase in the number of faults reported throughout the Auckland region following the storm.
"In response, Chorus has increased the number of technicians available to manage the workload. Technicians will prioritise medical escalations as they are raised by retail phone and broadband providers," the company said.
"Chorus apologises for any inconvenience and is working as quickly as possible to get households back up and running."
More than 160,000 households and businesses have been affected by the power outages. A total of 44,000 properties are still without power, many of which could take 'several days' to reconnect.
Many small businesses were also severely impacted by power outages.
A Z gas station located at a Northcross intersection in Auckland's North Shore had no power from 10pm on Tuesday and reconnected at 6am Wednesday morning.
Station manager Anirudh Hooda said the wild weather had severely impacted the business, which had lost approximately $5000 worth of fuel sales and $500 worth of pies, sausage rolls and hot food that had to be thrown out.
"Normally we have a morning rush, but we didn't this morning," Hooda said.
The entire Northcross area and the adjacent block of shops on the street had faced a lengthy power outage, he said.
"I was even asking customers and even they didn't have power at their home."
Hooda said it was a relief that customers had been understanding about the situation.
Taral Patel, the owner of the Price Cutter dairy next door, said he normally opened at 6am but without power was unable to make sales with the Eftpos machine down.
The power had come back on by 7.30am but by then he had lost an hour and a half of trading. "The first couple of hours are the busiest of the day."
Patel said the shop had lost power just before 9pm on Tuesday but fortunately the temperatures were low overnight so it wouldn't result in any fridge or freezer items having to be thrown out.
Patel said power cuts didn't happen very often so he didn't have a back-up generator for the shop.
He estimated he had probably lost around $750 to $800 worth of trading but said it would not be worth claiming on his insurance because the excess was $1000 alone.