Try using the below code in a simple console application first. Helps in isolating the issue :
List list = context.Web.Lists.GetByTitle("Pages");
var item = list.GetItemById(8);
item["PublishingPageContent"] = "<img src =\"/sites/test/PublishingImages/smiley.gif\" />";
Have modified it to use server-relative url of the image. Modify it as per your image location in SharePoint.
Also, have closed the image tag, but i dont think it going to affect end-result since its self-closing.