Senator Hanson and her former colleague had a bitter falling out after she reneged on a deal with the government to support company tax cuts.
Pauline Hanson, the leader of Australia's far-right One Nation party, wanted backing for her motion which stated "it is OK to be white".
Hanson's motion had been originally raised several weeks ago and had been on the Senate notice paper for some time.
"Yesterday, as a result of an administrative process failure, the government senators in the chamber ended up, on advice, voting in support of the motion", he said. Fair dinkum, Mathias. You're supposed to be better than this shit.
Attorney-general Christian Porter released a statement which said "an early email advising an approach on the motion went out from my office on this matter without my knowledge".
Mr Morrison was disappointed senators from this own side had backed a motion that borrowed a meme used by alt-right and nationalist groups in the USA and Europe and said it was "regrettable".
"It appears that, of the very large number of motions on which my office's views are routinely sought, this one was not escalated to me because it was interpreted in my office as a motion opposing racism".
'The associations of the language were not picked up.
Senator Hanson, who declared herself not to be a white supremacist, and fellow conservative crossbenchers didn't attend the recommitted vote.
"There are administrative processes in place to ensure that as a team we make the right decisions and on this occasion the process failed and I take responsibility for that", Cormann said.
Senator Cormann said when the One Nation motion first came up in September the government chose to oppose it and instead make a statement saying they deplored racism of any kind.
Senate President Scott Ryan gave the green light for Senator Burston to move a motion on Wednesday to refer her over pressing him to resign from the upper house.
"Looks like Scott Morrison is suffering from a bad case of it's okay to be white guilt", she tweeted on Tuesday.
However, not every member of the government walked back their support of the motion.