My training request expires in a week or so. Is it possible to extend it or do I have to submit a new one if I'm running out of time?
Assuming that you're in category 3 (aircraft under 12,500 lbs) then an approval is only valid for 365 days, so if you want to continue flight training after that then you need to request a new one:
Once you have permission to train, you have 180 days to begin training and 365 days to complete the approved training. Both of these time periods start from the day you receive approval. For example, if you begin flight training 30 days after you have been approved, you now have 335 days to finish. If you do not finish, a new training request must be submitted.