Well... it is telling the truth. You were late with a payment. OK, it was just one payment, and it was only a bit late.... But the report will show that - and usually it's clear as to how late - ie up to a month, up to two months etc.
It is not "such a bang", and as it becomes more distant, it will be less significant. Frankly alot of people miss the odd payment - it's normal and human. Miss a few payments? More than a month in arrears? Then you're talking trouble.
Barclaycard seem to have been spooked by it - but then you'd only just got the card. I really wouldn't worry...
A couple of years ago, I was the victim of a fraud. I had been overseas for 6 weeks and someone billed a visa card of mine £6800. The card had been in a locked filing cabinet in the UK all the time. In fact, it had only been used once since I got it. It was an "inside" job and the provider refunded without question. By the time I saw it, I'd had interest and late charges applied. This was refunded too - but they'd fumbled things a bit and sent me a bottle of wine (I never made an issue of it, just wanted the balance restored to zero). They said they'd reverse any detrimental information provided to reference agencies. Later they sent a letter confirming that in fact they'd never actually reported the late payments.
A year later I got copies of my credit reports. Guess what - late payments had been reported for two months on the trot. But it's never seemed to have affected my ability to get credit and I've never bothered to correct it. Well I have about 18 accounts showing (mostly settled, about 6 "live").
Things are tighter now, I know. But I really wouldn't worry and Jalexa has given you good advice. DD or not, you must keep up from now on (you know that, I know)!
I don't think you have any remedy against Barclaycard because of what the call centre person said.