First off, emission monitors are a bitch! There appear to be no standards for the emission monitoring routines implemented by different OEMs so the monitors appear to differ. That is why there is not a guaranteed single drive cycle that will get you 'there'. Generally there is no publicly available information for the monitors with the exception of a document that I discovered from Ford describing their monitor strategies.
Your description of the secondary air system pretty much describes what I experience on my 2000. Temperature does matter according to the service manual.
- the engine coolant temperature must be between 0C and 70C The 0C bit is confusing because I have started my car below freezing and I am pretty sure the air pump was running. Also, that kind of implies that fast heat up of the catalyst is not required when it is really cold which would seem like the time when you want it most.
- the engine coolant temperature has to have dropped by > 20C from the time that the engine was previously stopped
The running time is a maximum of 60 seconds; but, can be reduced based upon engine coolant temperature and intake air temperature. The SM provides no guidance on what those temperature sensitivities are. Note that these are the conditions for the secondary air system to run which is not the same as completion of the monitors.
There are a number of DTCs directly associated with the secondary air system.
P0410
P0411
P1410
P1415
P1416
Since these are later additions to the OBDII error code set, make sure that your scanner can actually report these error codes. Even if you do not have a CEL illumination, check for stored error codes. If you have any stored codes with respect to the secondary air system I expect the ECU will not execute the monitor routing. If you do have stored codes, clear them, cross your fingers and try a start where the engine matches the only two conditions that Honda has listed.
The Honda secondary air system is used to meet the LEV standard. The Ford document I have is for their LEV-II emission monitor and their system does not use a secondary air system to achieve compliance. The Honda secondary air system was a bit of a bandage. More modern engines use cam phasing as part of their cold start strategies. That said, I have screen grabbed the entry conditions for the Ford LEV-II monitor to complete.
View attachment 194084
Note that it refers to catalyst temperature. This is not actual catalyst temperature; but, an inferred catalyst temperature based on an algorithm which takes a number of other engine measurements into consideration. Also note that the minimum entry temperature for the monitor is 35 F which is a nudge above that 0C that the service manual notes so maybe that is the minimum temperature for the monitor to execute.
Successful completion of the Ford monitor is based upon the catalyst reaching a specified change in temperature (expressed as a temp at finish over temperature at start ratio) which is an inferred temperature. According to the Ford document the inputs to the inferred temperature calculation are engine speed, measured air mass (I assume they are a MAF engine) and commanded spark. Ford provides no guidance on how the algorithm uses this to calculate an inferred catalyst temperature. It would seem reasonable that since the objective of the secondary air system is to get the catalyst hot quickly that Honda might be using a similar inferred temperature calculation which means that failure to pass the monitor could be caused by an error or problem with one of those other inputs used in the inferred temperature calculation.
Assuming you have no stored DTCs relative to the secondary air system, are you letting the secondary air pump finish operation first before driving or are you driving off shortly after starting the car? I might try that as a variation and then checking the monitor status while the engine is still hot. As a Hale Mary you could clean the IAT because Honda's MAP system uses the manifold air temperature to infer air mass - that is assuming that Honda uses an inferred catalyst temperature for the air injection monitor Honda does state that the pump running time is a function of IAT so IAT problems could screw the system up.
That is my wild assed guess at what goes on with the secondary air monitor. All else fails, slaughter a chicken and sprinkle the blood on the car at mid night under a full moon to see if that helps.