I had a holiday in March which was completely ruined I tried

Hello, I'm an adviser with JA and I will try to help you.

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Hello, would you mind uploading the letter you received? (after redacting the personal information)

Can you clarify whether you declared that you were accepting the refund  from them "in full and final settlement"?

Can you also explain why you claimed the chargeback?

Hello?

I see.

And how much were you refunded in total (agent+chargeback) on a price of X that you paid for the holiday?

Also, when you claimed for the chargeback, did you disclose that you had got a partial refund from the merchant?

No worries for the call, it's offered by default by the platform.

So you had informed Barclaycard of the partial refund or were the informed by the travel agent?

When a chargeback is granted, the merchant can either accept it or challenge it.

In the latter case, the merchant will resubmit the charge along with a rebuttal letter and the necessary evidence to disprove the claim.

After this, the issuing bank will review the new evidence and make a decision and, if they find in favour of the merchant, the funds will be returned and an arbitration procedure is normally triggered. 

It is not clear whether, in this case, the travel agent challenged the chargeback or is trying to do now after missing the above described procedure of "representment".

However, since you received a letter threatening legal action, I must recommend that you speak with a solicitor, who will - presumably - dispute the right of the merchant to fight the chargeback at this point and in any event confirm that you had the right to the refund and compensation.

Diid you pay by credit card?

As you paid by credit card, your claim was classified as a Section 75 Consumer Credit Act 1974, for which the credit card company is jointly liable, hence the response from Barclaycard.

This should make your position stronger but, as I said, even more if the travel agent's claim is unlawful you need to get legal advice from a solicitor.