Is this a sentence fragment?
Hobbled across the court, leaning on the coach for support.

Yes!

Explanation:

To me, this feels very uncompleted, and sentence fragments are just that. Although it may look fine because of the comma, it is not. While reading it, you should get the feeling of incompletion. And that's what I got. We need a subject, or the main cause . And I think it doesn't have one.

To me there's just something missing. When identifying sentence fragments try to look for the three main components, which are the subject, the verb, and the thought.

I really hope this helped!

b. our party quickly made its way into the cave / but we failed to find our host [the cyclops] himself inside.