Hi JBgmat13,
To answer your immediate question: NO - taking the At-home GMAT a 2nd time will not 'replace' your 1st Score with your 2nd Score. Each Test-Taker has an 8-attempt limit, so you could potentially have up to 8 Official Scores on your Record (although there is currently a 2-attempt limit on the At-home version of the Exam).
GMAC has publicly stated that the Official Score that you earn on Test Day is within +/- 30 points of actual ability. Assuming a similar 'swing' in how your CATs function, your 4 CAT score results (which you noted here
https://gmatclub.com/forum/how-to-continue-354405.html) - along with your Official Score - show that you essentially performed the same each time (about 630 +/- a few points). You handle certain aspects of the GMAT consistently well, but you also make certain consistent mistakes. With this 610, you're closer to a 650+ than you probably realize, but you will have to make some fundamental changes to how you 'see' (and respond to) the GMAT before you will be able to consistently score at that higher level.
If you have not made any definitive changes to how you work through the GMAT, then you might not want to 'rush in' to retake the Exam too soon (since you will likely Score in this same general range). Before I can offer you the specific advice that you’re looking for, it would help if you could provide a bit more information on how you've been studying and your goals:
1) What were your Quant and Verbal Scaled Scores on your Official GMAT?
2) How long have you studied? How many hours do you typically study each week?
3) What study materials have you used so far?
4) On what dates (or approximate dates) did you take EACH of your CATs/mocks and how did you score on EACH (including the Quant and Verbal Scaled Scores for EACH)?
5) When are you planning to apply to Business School and what Schools are you planning to apply to?
GMAT assassins aren't born, they're made,
Rich