Poor road conditions arising from the freezing weather have created difficulties in getting money to post offices in some areas, An Post has said.
Sinn Féin leader Mary Lou McDonald had earlier claimed that some post offices were running out of money due to the doubling of the fuel allowance payment, a move taken to help people keep their homes warm during the cold snap.
Ms McDonald said she had received reports from Dublin, Dundalk and Tullamore of cash shortages.
However, An Post said 1,000 post offices had opened on Thursday and that less than 10 were affected by a shortage of cash.
Ms McHugh denied this was connected to the fuel allowance payments insisting it was due to the treacherous driving conditions.
She said the company regretted any inconvenience caused to customers but that people expecting payments would receive them.
Social welfare payments due for collection will be available when post offices open again, hopefully on Saturday, she added.
The Government announced on Wednesday that recipients of the fuel allowance would receive an additional €22.50 this week due to the poor weather.
Minister for Social Protection Regina Doherty confirmed some of the money would not be paid until next week.
Ms McDonald said this could result in those using pre-paid meters having to be cautious with their energy use.
Fianna Fáil TD Timmy Dooley also questioned how An Post ran out of cash and ATMs have not.
Mr Dooley said it was “not good enough that Irish people and businesses are affected in this way”.
