Does PGR have auto-combat?

No, the game doesn't have auto-combat. However, most of the battles are quite short, lasting around 1-2 minutes so auto-combat seems to be unnecessary. The game however, has an Auto-Clear option which allows you to clear previously-cleared stages automatically.

Auto-clearing will run in the background and after the timer has run out, you can collect the rewards. You can do multiple auto-clearing on the same stage, at the cost of higher stamina and longer completion time.