Hyaluronic acid does not cause gout. Uric acid buildup in the joints does.
It is believed that HA exists in the body independently when gout is present, according to a study, but nothing in that study indicated it has anything to do with causing it.
Hyaluronic acid shots are being used to treat osteoarthritis in the knee, as it offers more lubrication for joints. It would follow that using the same shots on gout might be helpful, but this has not yet happened.