Q: Can the secondary air injection relay cause my transmission to randomly not perform as it should?

Since the majority of today’s (if not all of them) are now controlled by electronic transmission control modules that receive inputs from the ECM, any glitch in the functioning of the motor may potentially cause a domino effect resulting in the transmission potentially not shifting properly. When the transmission control module receives inputs from the ECM, it looks at a number of things, one of which is RPM’s achieved at specific intervals. When this does not happen, this may cause a delay in proper shifting intervals as you describe. There are many different things that impact air/fuel and spark in the motor which all result in proper firing of the motor. One of these is also the secondary air injection pump since it pumps additional air into the motor in order to perform its function. When this is not working properly, this can impact the proper fuel/air ratio or tune of the motor resulting in the check engine light coming on and other performance problems. I would suggest having a professional from Yourmechanic coming to your location to take a look at the vehicle.