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Would it be an issue if I were to send a 'statement of earnings' for tax year 16/17 instead of a P60. My employer won't send the P60's out until the end of the month and I've got 10 working days to send my PR application (Online). I've got the other 4 p60's with my application.

or would that delay my application

No, that's fine. Just explain in a cover letter about your situation.
'statement of earnings' will be signed and dated there should be no issue with that.