You can do whatever you like with it: leave it alone, delete it, change its status to something else. Whatever.
No, you can't re-try the payment, at least not from your admin area.
Alternatively you could login to your PayPal account and use the "Request Money" option, and send them a request for payment. Then when they complete the payment, you could manually go into that order in your admin and update its status to Processing or Shipped, and add a comment indicating the new PayPal transaction information you receive from PayPal. That's the only way to "link" the new payment to the old order.
Or have them place a new order.
.
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