I don't know what a phrasal is, so let me start with the basics:
Notice first that the word is around, not round. (However, in conversation, sometimes people shorten around to 'round, with an apostrophe.)
Around is a preposition, like over, under, on, and off. It can mean surrounding or near.
The use you're asking about is an idiom. Getting around to something is like getting to something. It strongly implies that it could easily have been done if it had been more important to the person.
She had all evening, but she never got around to her homework.
A week later, he still hadn't gotten around to washing his hair.
So the article is telling you that Burris did have time to turn in his gun, but neglected to it anyway - maybe because he's lazy, forgetful, or in love with his gun.
[I edit copy and have tutored college writing.]
Student or Learner