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Conservative finance spokesman Andrew Johnston said Glasgow had secured more funding because it administered homelessness prevention through the city's integration joint board, which is in charge of health and social care, while in Edinburgh it was administered directly by the council.
He said: "Last year Glasgow was able to recover £8.8m through the 2020/21 health mobilisation plan arrangements, while Edinburgh was unable to recover equivalent costs of £9.3m."
Cllr Johnston said: “It seems that due to administrative decisions taken some years ago Edinburgh is losing out in terms of money for homelessness.
“I think we've got an opportunity to make the case that we should be allocated more money and we shouldn't lose out just because we deal with something in a slightly different bureaucratic way. I believe an all-party approach is likely to have the highest chance of success.
“Unfortunately homelessess is still a real problem in Edinburgh and any extra money we can get to help would be much appreciated and if it's well targeted could really make a difference.”
He said he wanted the Scottish Government to recognise the anomaly and give Edinburgh the appropriate funding, but if it was necessary to switch homelessness prevention to the Edinburgh IJB in order to get the money that should be discussed.
"I think we do it the right way through the council because I think there's more scrutiny.”
Cllr Munn said he welcomed Cllr Johnston's letter.
"He makes a good point and it is something we have been lobbying the Scottish Government on for some time.
"I welcome Cllr Johnston’s offer to work together on the issue. I’m always in favour of, where we can, working across party groups to help the situation in Edinburgh.”
He said the situation had been highlighted by council officers.
“It's a different set-up in Glasgow and Edinburgh and arguably it puts Edinburgh at a disadvantage at a time of great pressure. It's to do with the different ways mobilisation money was allocated and it has thrown up this anomaly.
"Covid has exacerbated the homelessness issues in Edinburgh, so it's something we would like to see being addressed.”
Cllr Munn said there had “not yet” been a positive response from the government.
“We are in regular discussion with the Scottish Government on a range of budget issues and homelessness is one of the issues which is pressing in Edinburgh.
"There was clear pressure before Covid hit, but Covid has made it worse so it's something raised regularly by the council leader and other spokespeople.
“This is something we'll keep lobbying on.”
He said extra funding would help the council to address the issue, particularly the problem of temporary accommodation.
“Once people find themselves homeless you want to get them into permanent accommodation as quickly as you can and at the moment that's very difficult in Edinburgh.
“There's a lot of work being done by council staff to try and transform the service but it's a very difficult time and pressure is just ramping up all the time.”