Sinn Fein Foyle MLA Martina Anderson
The senior MLA and convicted IRA bomber had been criticised for making the remarks on social media on Tuesday.
Her tweet was later deleted.
In a statement, Ms Anderson said her comments had been “clumsy” but were not directed at victims.
The Foyle MLA and former MEP also claimed the money was mainly for those who took part in what she described as “Britain’s dirty war in Ireland”, and would mostly go to “those involved in collusion” and British troops, for instance paratroopers involved in shootings in Ballymurphy in 1971 and on Bloody Sunday in 1972.
“It was never my intention to cause them any hurt,” she said on Wednesday.
“All victims of the conflict deserve acknowledgement of their pain and loss and I support them in their efforts to get their pension.”
Sinn Féin President Mary Lou McDonald said it was right that Ms Anderson had apologised.
She told Irish national broadcaster RTÉ that her party still had concerns about the pension scheme, but that the comments had caused hurt to victims and an apology was “absolutely necessary”.
“All of us recognise what they’ve been through they are absolutely entitled to their pension,” she said.
Earlier, a woman who challenged the delay in a compensation scheme for Troubles victims described Ms Anderson’s tweet as “very hurtful and insulting”.
Jennifer McNern, who lost lost both legs in an IRA bomb attack on the Abercorn Restaurant in 1972, said the payments were for “forgotten victims”.
“We as a group have campaigned for over 10 years for people who have been injured through no fault of their own, people who have been forgotten about and pushed aside the whole way through the peace process,” she said.
“If you see the people who are applying for this pension, they are blind, they are paralysed and they are amputees.
“That is the people who will avail of this pension when it opens.”
Last week, a judge ruled that the NI Executive Office had acted unlawfully in delaying the scheme, which was approved by Westminster in January.
It followed a number of legal challenges brought against the delay, one of which was taken by Ms McNern.
SDLP leader Colum Eastwood
“These comments, coming less than a week after the High Court criticised the Executive Office, and specifically the deputy first minister, for failing to meet their obligations to these victims under the law, are astonishing,” he said.
“Time and again, Sinn Féin has proven that it has no interest in putting the needs of victims first.”
DUP Foyle assembly member Gary Middleton described Ms Anderson’s remarks as “a deeply offensive tweet about innocent victims”.
“This cannot be allowed to go unchecked,” he continued.
“People who have lived most of their lives with shrapnel from an explosion in their body or who are haunted with the smell, taste and noise of a bombing should not be labelled by Martina Anderson.”
Mr Middleton added he would raise the comments with the assembly authorities as to whether it breached the MLAs’ code of conduct.
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