Do you mean "worried, gaze-averting children" (children who are worried and look away) or "worried-gaze-averting children" (children who avoid worried gazes)? It's closer to the former: some children are staring off into space seemingly worried (because they don't know what to write).
The second part means that it's evident that こうへい is no just trying, but is already (extremely) motivated to write neatly: the characters he's written down so far show that he's really doing his best.