Two Conservative councillors left a key meeting after members voted for it to go ahead during the period of national mourning.
The party's group leader said it was "not the time" for policy boards to continue, following the passing of Queen Elizabeth II last week.
Councillor Neill Graham moved for the Leadership Board to be adjourned until the conclusion of national mourning, which will be at the end of the day of the State Funeral on Monday.
However, just he and his depute Councillor Alec Leishman voted in favour of a delay as 13 councillors opted for business to continue.
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SNP council leader Iain Nicolson and Labour group leader Iain McMillan cited the importance of one item in particular, which included considering a series of relief measures in response to the cost-of-living crisis.
Those proposals, which includes winter clothing payments and plans for warm spaces, were given the green light at the meeting.
At the board's outset, Councillor Graham said: "Due to the death of Her Majesty Queen Elizabeth II, a period of national mourning is now in place.
"Due to today being in that period of mourning and in line with other councils and parliament, we here in Renfrewshire should respect this period.
"This is not the time for policy boards to continue and I'd like to move that this board is adjourned until after the period of national mourning is over."
Councillor McMillan was sympathetic towards the view of his counterpart, adding that he had thought "long and hard" about the situation.
However, he felt the board needed to go ahead because of the item in question, which would see around half a million pounds worth of initiatives approved as rising inflation and energy costs inflict several financial pressure on people in the area.
He said: "It requires action from us today and that's to get money out to the most vulnerable in our communities.
"That's the reason why the Labour group is quite happy to continue with the meeting today.
"Again, I emphasise I do have some sympathy with Councillor Graham's motion."
Councillor Nicolson said that the council was operating within the guidelines of Operation Unicorn, the codename for the plan for handling the Queen's death in Scotland.
He added: "I'm completely in agreement with your view, Councillor McMillan, about some of the business on the agenda.
"It's stuff that we've been working to with regards to other things that are going on in people's lives and they're quite important."
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