Not an expert, have been looking around and does look like canker, the colour of that branch also doesn't look good or perhaps the photo.
If this were a canker most controls indicate removal and burning infected branches and then sealing wounds. How critical are these branches to the design, if you can manage without them might be safer to remove. As mentioned above treat with fungicide as well, as this could be the cause of the canker, although here it seems only works for some fungi.
If in doubt try and take to local nursery and ask another opinion, see if any others respond.
Mites won't make the branches weep but they can kill individual branches. I'm not sure about Sydney but here mites are not really active in winter. They prefer dry weather.
Scale is also worth considering. Cypress canker does attack junipers and one of the symptoms can be weeping sap here and there. I agree that complete removal of the affected branch down to healthy tissue is necessary together with sealing the cut and spraying the entire tree with copper oxychloride and diluted Lime Sulphur a few times. There's no guarantee of recovery from Cypress canker.