In a TV address, Park Geun-hye apologized for allowing Choi Soon-sil, a long-standing friend of her, inappropriate access to government policy-making.
The president agreed to be questioned over the scandal but did not offer to resign.
Choi Soon-sil is suspected of using her friendship with the president to solicit donations to a non-profit fund she controlled.
She is now in detention facing charges of fraud and abuse of power.
The main opposition party said Park Geun-hye’s apology lacked sincerity and it called on her to step back from state affairs.
Scuffles broke out between police and demonstrators demanding Park Geun-hye’s resignation in central Seoul on November 4.
Choi Soon-sil is the daughter of Choi Tae-min, a shadowy quasi-religious leader who was closely linked to Park Geun-hye’s father, then-president Park Chung-hee.
On November 4, Park Geun-hye went on TV to deny allowing cultish rituals to be held in the presidential palace.
“There have been claims that I fell for a religious cult or had [shamanist rituals] performed in the Blue House, but I would like to clarify that those are absolutely not true,” she said.
The president said she took sole responsibility for access to government documents and was willing to be investigated.
She had “put too much faith in a personal relationship and didn’t look carefully at what was happening”.
Anyone found to have done wrong would be punished, Park Geun-hye said, and “if necessary, I’m determined to let prosecutors investigate me and accept an investigation by an independent counsel too”.
The scandal has left the president with an approval rating of just 5%.
Park Geun-hye has already replaced her prime minster, reshuffled her cabinet and dismissed several aides, but there are growing calls for her resignation or impeachment.
Choo Mi-ae, leader of the main opposition Democratic Party, said Park Geun-hye did not believe the apology was genuine and called on her to accept a new prime minister recommended by parliament.
Park Geun-hye became South Korea’s first female president when she was elected in a close-run contest in December 2012.
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