I am going running. I am going on a run.

The usual form is “I go running [swimming, hiking, camping, ...]. This explains what a person did, does, or will do. Example: What do you do in your free time? I go running most days and I go camping in the summer. A less common form is “I go to place X to run [swim, hike, camp, ...].” This explains why a person goes to a place. Example: I go to the park to relax a bit and feed the pigeons. Me, I go [to the park] to run.

I go running on morning. I go run on morning .