The charity has handled 2,500 counselling sessions in the last three years while centres in Glasgow and Aberdeen have dealt with 159 calls on the subject from children in the last year.
Chancellor: Scots Should Not Have Different EU Deal
The UK's new Chancellor has said Scotland should not have a different relationship with Europe than the rest of the UK.
Philip Hammond, who replaced George Osborne as new Prime Minister Theresa May appointed her Cabinet, said the vote to leave the European Union was a "democratic decision'' made by the United Kingdom as a whole, which would now be implemented.
It marks a blow for First Minister Nicola Sturgeon, who has been consistently making a case for Scotland to retain its links with Europe after the majority of voters north of the border opted for Remain.
Her Scottish Government is already working on legislation that would be needed for a second independence referendum to take place, in case ministers feel leaving the UK is the only way to keep Scotland in the EU.
But Mr Hammond says the best future for Scotland is remaining "inside the United Kingdom economy''.
Asked if he could envisage a situation where Scotland has a different relationship than the rest of the UK with the EU, the Chancellor said: "No.''
He added: "I think the best future for Scotland is inside the United Kingdom economy, let's make this United Kingdom economy work for all of us, and let's negotiate with the European Union, from outside the European Union, a relationship which works for Britain and works for Europe, so we can have as close a relationship in trade and commerce as we possibly can while being outside the European Union, as the British people have determined we should be.
"However we voted, we are part of the United Kingdom and we have democratic decisions made across the United Kingdom. We will now implement the decision that the people of the United Kingdom collectively have made to leave the European Union.
"But that does not mean turning our backs on Europe, it does not mean closing our doors to our neighbours and partners in Europe, it means working as closely as we can with them to maximise economic co-operation, trade and business opportunities without being a member of the political construct of the European Union.''
Mr Hammond pledged the new Government led by Mrs May will "prioritise stability in the economy'', adding that "whatever we need to do in order to deliver that stability, we will do''.
He also insisted the Barnett formula, used to determine the amount of funding that goes to Scotland and other parts of the UK, would stay in place.
"There are no plans to change the Barnett formula,'' Mr Hammond said.
"The Barnett formula works, it's an arrangement that works well, it has worked well for many years. Like many things in our constitutional arrangements it has worked rather well, it's a pragmatic solution and there are no plans to change it.''
Nicola Sturgeon described the Chancellor's comments as "deeply disappointing''.
She said: "I have been absolutely clear on this issue - the people of Scotland voted decisively to stay part of the European Union and their wishes must be respected.
"That includes respect from the UK Government, which is why Philip Hammond's comments are deeply disappointing - I very much hope the new Prime Minister will be more open to constructive discussion.''
She added: "The Scottish Government is pursuing every possible avenue to protect our place in Europe - which of course means protecting businesses' freedom to trade, the ability of workers to be protected, and our right to continue to influence EU decisions.
"That work continues today with the first meeting of the standing council of experts I have assembled. We will continue to explore all options to protect Scotland's place in Europe - and I have made clear that the option of an independence referendum must be on the table if it becomes clear that that is the best or only way of preserving our EU status.
"I hope that the new PM, Mr Hammond and all of the UK Government will understand that we are absolutely serious when it comes to achieving our goal of protecting Scotland's vital interests.''
Official figures show more than 400 consultant jobs across Scotland are unfilled.
Education Secretary John Swinney will urge rival parties at Holyrood to back his plans for sweeping reforms to Scotland's schools.
Experts described the number as a "real concern''.
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