As long as your address bar shows the correct address for your bank or council (starting with 'https' to show that it's a secure site) you've got absolutely nothing to worry about.
The problem comes about because of some sloppy coding by the people who designed those websites. For example, lets assume that the page you're viewing is
https://mycouncil.gov.uk and includes an image called taxpayer.jpg. Your browser looks at where the image is loaded from and expects to see it in a secure location, such as
https://mycouncil.gov.uk/images/taxpayer.jpg. As long as it does, the padlock will remain as normal.
But a lazy website designer might call up that image from a folder, within the site, simply called ./images/taxpayer.jpg. The browser spots that the 'https://' bit (which identifies a secure site) is missing and so flags up an unnecessary warning about the presence of unauthenticated content.
Chris