I have an anchor that changes its background image when hovered with a class class-btn that contains a background-image.
When hovered, it
The easiest way to avoid this is to make use of image sprites. For a good overview, check out this CSS Tricks article.
That way, you not only solve the flicker problem you're seeing, but will also reduce the number of HTTP requests. Your CSS will look something like:
a.class-btn { background: url('path/to/image.jpg') 0 0 no-repeat; }
a.class-btn:hover { background-position: 0 -40px; }
The specifics will depend on your images. You can also make use of an online sprite generator to make the process easier.