[CQ-Contest] Questions on Pile-up Management

[VA3NR]

I have a couple questions for those experienced running CW pile-ups:

1) I know its best to pick out a full callsign after just one call.  My question is for those times when you can't pick out a full callsign, is it better to send a partial call (e.g. 1AW 5NN4) to keep the rhythm going or should you pause and allow the pile-up to call second a time?

2) Occasionally when I get the chance to run a CW pile-up I'll get callers who seem to mis-read my exhange to someone else as their own.  For example, I send  W1AW 5NN4  and besides the intended  W1AW  I also get some other station sending me a report.  If I just send  R TU  to continue the pileup then both  W1AW plus the mystery caller are logging me but I'm only logging W1AW.  The mystery caller isn't going to call me again in the test because he thinks we've worked and I've lost a potential QSO.  Is there a good way to recover these QSO's without breaking rhythm?

Thanks,
Chris VA3NR.