How can I fill in the gap between my chain-link fence and its gate
If you don't care about how it looks, a piece of plywood or other rigid material mounted to the gate would bridge the gap.  You could attach it to the vertical part of the gate using a variety of types of u-bolts/brackets.

Otherwise the easiest thing is to replace the gate with a wider one. Another alternative is to move the posts to reduce the gap on the existing gate but this requires installing posts and replacing the section of fence.