Am I right in thinking that I can claim back the tax on the first £6,030 I earnt, or do I waive that by having earnt above my threshold?
Provided you pass your tax details on (P45 or whatever) when you changed jobs then it would have all been taken care of automatically. From the figures you quoted I'd say you were in effect re-imbursed the over payment by having less tax deducted from your current job.
And as Talk Talk said, they sort any minor discrepancies out with your tax code the following year.