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Which Barclay card are you after?
If you have a recent late on your Transunion, I would hold off applying for any TU puller (Barclay, Capital One) until you get that resolved.
You can apply all in one day or spread it out in days, weeks. You'll get hard pull inquiries regardless.
I would agree with the others in the order below.
1. PenFed
2. Barclay
3. Capital One.