Only a Carrier dealer can pinpoint the problem, and he will need the model number and serial number. Meanwhile, there is apperantly a flame sensor that is on the blink. It doesn't see the flame most of the time so it shuts the power down (it doesn't want to fill the room with gas). A possible problem with the sensor is, it may be covered with soot and can't "see" the flame. One thing you could do in the interim is turn your fan to the "on" position on the thermostat. This will keep the air circulating throughout the house and use the little bit of heat created by the five second burns, this will leave only the flame to cycle.
My fan has been on for 11 years and my heat pump compressor cycles as necessary for heat or cooling.