There is a 6 minute delay from when my heat pump compressor comes on until the inside blower comes on. It used to only delay a few seconds. This is a Goodman A60-20 air handler. Do I need to replace the blower time delay relay?

I would just eliminate it. Just hook common and "G" directly to the fan relay and that will fix it. It should only be a few seconds, but basically it's a useless feature. Normally, the time delay is wired to run the fan a few minutes after the compressor turns off to get the chilled air out of the unit.