The on-line NSX service manual indicates that P1607 is 'Engine Control Module Internal Circuit Failure A'. The manual does not list a P1410 error code at all for the NSX. The error codes related to the EGR valve are P0401, P1491 and P1498. I believe the codes that start P1xxx are Honda specific codes in which case a generic code reader might not interpret them correctly.
P1607 sounds ominous; however, ECM problems can be created by supply voltage problems (usually low volatage). Obvious places to look are bad battery, bad battery connections and problems with the main relay. It might be possible that you have low voltage on initial start-up because of a bad connection but then once running, the alternator/voltage regulator is maintaining an adequate system voltage for the ECM. I recall something about the ECM having to trap an error 3 times before the ECM triggers the MIL. After you clear the codes, you might get two re starts without triggering the MIL. On the third re start, you would trigger the MIL - if the problem is caused by low voltage during start up. I don't know for sure that low voltage is your problem; however, you want to rule it out as a possible cause (primarily because the other sources of the problem are much more expensive). Also, the fact that you have not identified any other operating issues with the engine suggests that the ECM proper may be OK and that the P1607 code is a symptom resulting from some other problem.
It would be worthwhile to check out the operation of your EGR valve. In particular the valve lift sensing circuit. If you have a problem with the valve lift potentiometer or the wiring from the sensor to the ECM, you could be buggering up the EGR lift signal back to the ECM and you could be depressing the voltages within ECM if there was an internal short within the valve lift sensor or a short to ground within the valve lift sensor. These shorts could also be in the wiring between the ECM and the EGR sensor. If you can drive the car for an extended period of time without throwing a code related to EGR operation, then I am going to guesss that the problem is probably not EGR related; however, it is worth eliminating as a possibility. Unfortunately, I could not find the EGR lift sensor test procedure in the on-line manual and my paper manual is in the trunk of my NSX which is wrapped up in a secure storage facility for the winter. Others may be able to give you advice on the EGR test procedure.
As I noted before, I don't know for sure that low voltage on start up is the cause of your problem; however, it is a possible cause and something that you want to eliminate as a possibility first because everything else starts to cost $$$$. Good luck with it!