Hi cheuna105,
To start, a 610 is a strong initial CAT score (the average score on the Official GMAT hovers around 550 most years). That having been said, many GMATers spend 3 months (or more) of consistent study time before they hit their 'peak' scores, so if your studies are going to be limited to just 4 weeks, then there will almost certainly be a limit to how much you can improve - and you will likely need to take the At-home version of the GMAT.
A number of people who have taken the At-home GMAT have found it more challenging than they originally expected (among the various issues, using the online notepad is considerably different from taking notes by hand). Currently, you're only allowed to take the At-home GMAT just ONCE, so if you have any interest in applying to any OTHER Schools, then 'rushing in' to take the At-home Exam - and potentially scoring lower than your Goal Score - might not be the best choice.
You might consider reaching out directly to this School to see how late you could submit your GMAT Score (assuming that you submitted the rest of your application on time). The School's representatives might give you a little leeway on submitting your Score - and even a couple of additional weeks of study time could potentially help you a great deal under these circumstances.
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:
Studies:
1) What study materials have you used so far?
2) Are you interested in any other Schools? If you did not receive an invite to this one School, would you not apply anywhere else (or perhaps apply again to this School later on)?
3) What is the average GMAT School for the specific Program that you're interested in?
GMAT assassins aren't born, they're made,
Rich