"Great intentions" refers to having good or positive motives or plans for doing something. It means having the desire to do the right thing, even if the outcome may not always be successful. Full definition
Option Two: you could think with greater intention about why you are overwhelmed by your emotions. (christianitytoday.com)
I had such great intentions of making these last year and well... didn't. (wingitvegan.com)
You're dropping your kids off at your ex's with great intentions about giving your kids two good homes and sharing the parenting responsibilities. (care.com)