Can I Take My Puppy Running With Me?

I have a 5month old puppy. he is 12 lbs and shouldn’t get much bigger than 15lbs (he is already bigger than both his parents which are a havanese and a maltese).
I am thinking on taking him on 2-3 3km (1.8mile) runs with me as i begin training for a 1/2 marathon. I am very slow, he doesn’t get to more than a “trot” pace himself (i run at a pace of 6min/km – these runs take me about 18-20min).
I’ve heard it’s not good to “run” with a puppy. but would this be considered “running”?? he gets much harder workouts playing fetch in the backyard, but at the same time i would feel horrible if it turned out i was hurting him!!
thanks for all advice!
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